Archimedean Property
Definition

$\forall ϵ > 0, \exists n \in N, \frac{1}{n} < ϵ$

For any positive real number $ϵ$ , there exists a positive integer $n$ such that $\frac{1}{n}$ is less than or equal to $ϵ$

Proof

Let $\forall n \in N, \frac{1}{n} \geq ϵ$ , then $\frac{1}{ϵ} \geq n$ Since $N$ is not bounded above, It is a contradiction
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Dense Set
Definition

a subset $A$ of a Topological Space $(X, T)$ is dense in $X$ if the Closure of $A$ in $X$ is equal to $X .$ $\overset{ˉ}{A} = X$

Or, equivalently, for every element of $X$ , every Neighborhood of $x$ intersects $A$ . $\forall x \in X, \forall U := N_{x}, U \cap A \neq = \emptyset$

Denseness of the Rational Numbers

$\forall x, y \in R, a < b \Rightarrow \exists r \in Q, a < r < b$

Denseness of the Irrational Numbers

$\forall x, y \in R, a < b \Rightarrow \exists s \in I, a < s < b$
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Real Number
Definition

$R$

Construction

Construction of the Real Numbers

Facts

Law of Trichotomy
Definition

$\forall a, b \in R, a > b or a = b or a < b$ Every Real Number is either positive, negative, or zero.
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When $s \in R$ is an Upper Bound of $S$ , the following are equivalence relations.

$s = sup (S)$

$\forall ϵ > 0, \exists x \in S, s - ϵ < x \leq s$

$\forall ϵ > 0, S \cap (s - ϵ, s] \neq = \emptyset$

Every closed interval in $R$ is uncountable (by the Theorem)

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Field
Definition

An algebraic structure consisting of a set $R$ equipped with two binomial operations: addition( $+$ ) and multiplication( $\cdot$ ) such that the following conditions hold.

$R$ is an Abelian Group under addition.

$R$ is Abelian Group under multiplication.

Multiplication is distributive with respect to addition. $\forall r, s, t \in R : [r \cdot (s + t) = (r \cdot s) + (r \cdot t)] \land [(s + t) \cdot r = (s \cdot r) + (t \cdot r)]$

A commutative division Ring
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Order Axiom
Definition

Totally Ordered Set

$\forall x, \forall y, \forall z \in R, x < y \Rightarrow x + z < y + z$

$\forall x, \forall y, \forall z \in R, (x < y) \land (z > 0) \Rightarrow x z < yz$

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Completeness Axioms
Definition

$\forall x \in S \subset R, \exists L \in R, x \leq L$ Every non-empty subset of Real Number having an Upper Bound must have a Supremum

Facts

Rational Number does not satisfy completeness axioms

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Construction of the Real Numbers
Definition

Axiomatic Definition

Let $R$ denote the set of all real numbers, then

The set $R$ is a Field

The set $R$ satisfies Order Axiom

The set $R$ satisfies Completeness Axioms

Construction from Cauchy Sequences

Let $R$ be the set of Cauchy Sequence of rational numbers

$(a_{n}) \sim (b_{n}) \Leftrightarrow n \to \infty lim (a_{n} - b_{n}) = 0$ Define an Equivalence Relation $\sim$ on the $R$ The two cauchy sequences $(a_{n}), (b_{n})$ are equivalent if the sequence $(a_{n} - b_{n})$ converges to $0$ .

The set of real numbers, denoted by $R$ , is defined as the set of equivalence class under the relation $\sim$

$(n \to \infty lim a_{n} = α) \Rightarrow ([(a_{n})] := α)$ and denote the value represented by the Equivalence Class as a limit of the Cauchy Sequence

The operations of addition and multiplication on $R$ is defined as follows

$[(a_{n})] + [(b_{n})] = [(a_{n}) + (b_{n})]$

$[(a_{n})] \cdot [(b_{n})] = [(a_{n}) \cdot (b_{n})]$

Construction by Dedekind Cuts

Let $C$ be the subset of the set of Rational Number $Q$ If $C$ fulfills the following conditions, then $C$ is a real number

$C \neq = \emptyset$

$\forall s, t \in C, t < s \Rightarrow (s \in C \Rightarrow t \in C)$

$∄ u \in C, \forall s \in C, s \leq u$

$\forall c \in C, \exists x \in Q, c < x$ or $C \neq = Q$

By the conditions, we have properties $\forall s \in / C, ∄ t \in C, t > s$ $\forall r \in C, \forall s \in / C, r < s$

We define a Total Order Relation on the real numbers as follows, $x \leq y \Leftrightarrow x \subset y$

Operations

Let $A, B \in R$

$A + B := {a + b ∣ a \in A \land b \in B}$

$A - B := {a - b ∣ a \in A \land b \in (Q ∖ B)}$

$- B := {a - b ∣ a < 0 \land b \in (Q ∖ B)}$

If $A, B \geq 0$ , $A \times B := {a \times b ∣ a \in A \land b \in B}$

If $A \geq 0, B \leq 0$ , then $A \times B := - (A \times (- B))$

If $A \leq 0, B \geq 0$ , then $A \times B := - ((- A) \times B)$

If $A, B \leq 0$ , then $A \times B := (- A) \times (- B)$

$\frac{1}{B} := {\frac{a}{b} ∣ a < 1 \land b \in (Q ∖ B)}$

$A \div B = \frac{A}{B} := A \times \frac{1}{B}$

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Limit of a Function
Definition

Function of a Single Variable

Epsilon-Delta-Definition of Limit

Suppose $f : D \to R, a \in D, L \in R$ $\forall ϵ > 0, \exists δ > 0, \forall x \in D, (0 < ∣ x - a ∣ < δ \Rightarrow ∣ f (x) - L ∣ < ϵ) \Leftrightarrow x \to a lim f (x) = L$

The limit of $f$ , as $x$ approaches $a$ , is $L$ , or say the $f$ converges to $L$ , otherwise, $f$ diverges

Right-Sided Limit

Suppose $f : D \to R, a \in D, L \in R$ $\forall ϵ > 0, \exists δ > 0, \forall x \in D, (0 < x - a < δ \Rightarrow ∣ f (x) - L ∣ < ϵ) \Leftrightarrow x \to a^{+} lim f (x) = f (a^{+}) = L$

The right-sided limit of $f$ , as $x$ approaches $a$ from the right side, is $L$

Left-Sided Limit

Suppose $f : D \to R, a \in D, L \in R$ $\forall ϵ > 0, \exists δ > 0, \forall x \in D, (0 < a - x < δ \Rightarrow ∣ f (x) - L ∣ < ϵ) \Leftrightarrow x \to a^{-} lim f (x) = f (a^{-}) = L$

The left-sided limit of $f$ , as $x$ approaches $a$ from the left side, is $L$

Limits at infinity

Suppose $f : R \to R$ and $a, L \in R$ $\forall ϵ > 0, \exists N \in R, \forall x \in R, (x \geq N \Rightarrow ∣ f (x) - L ∣ < ϵ) \Leftrightarrow x \to \infty lim f (x) = L$

Infinite Limits

Suppose $f : R \to R$ and $a, L \in R$ $\forall M > 0, \exists δ > 0, \forall x \in R, (0 < ∣ x - a ∣ < δ \Rightarrow f (x) > M) \Leftrightarrow x \to a lim f (x) = \infty$

Suppose $f : R \to R$ and $a, L \in R$ $\forall M > 0, \exists N \in R, \forall x \in R, (x \geq N \Rightarrow f (x) > M) \Leftrightarrow x \to \infty lim f (x) = \infty$

Function of Multi Variables

Ordinary Limits

Suppose $f : D \subset R^{n} \to R^{m}, a \in D, L \in R^{m}$ $\forall ϵ > 0, \exists δ > 0, \forall x \in D, (0 < ∣∣ x - a ∣∣ < δ \Rightarrow ∣∣ f (x) - L ∣∣ < ϵ) \Leftrightarrow x \to a lim f (x) = L$ where $∣∣ x - a ∣∣$ denotes the Euclidean distance between $x$ and $a$ in $R^{n}$ .

Properties

Algebraic Limit Theorem

Suppose $f, g : D \subset R^{n} \to R^{m}$ , and $a \in D$ Then, the following are satisfied

$x \to a lim f (x) + g (x) = x \to a lim f (x) + x \to a lim g (x)$

$x \to a lim f (x) - g (x) = x \to a lim f (x) - x \to a lim g (x)$

$x \to a lim f (x) \cdot g (x) = x \to a lim f (x) \cdot x \to a lim g (x)$

$x \to a lim \frac{f ( x )}{g ( x )} = \frac{x \to a lim f ( x )}{x \to a lim g ( x )}$ where $x \to a lim g (x) \neq = 0$

$x \to a lim [f (x)]^{n} = [x \to a lim f (x)]^{n}$ where $n \in N$

$x \to a lim [f (x)]^{1/ n} = [x \to a lim f (x)]^{1/ n}$ where $n \in N$

Facts

Suppose $f : D \to R$ , and $a \in D$ If $x \to a lim f (x)$ converges, then the limit is unique

Squeeze Theorem for Functions

Suppose $f, g, h : D \to R$ , $L \in R$ , and $\forall x \in D - {a}, f (x) \leq g (x) \leq h (x)$

If $x \to a lim f (x) = x \to a lim h (x) = L$ , then $x \to a lim g (x) = L$
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Continuous Function
Definition

Topological Definition of Continuity

Consider two topological spaces $(X, T_{X}, B_{X})$ and $(Y, T_{Y}, B_{Y})$ . A function $f : X \to Y$ is continuous if
$\forall V \in \mathcal{T}_{Y},\ f^{-1}(V) \in \mathcal{T}_{X}\ &\Longleftrightarrow\ \forall B \in \mathcal{B}_{Y},\ f^{-1}(B) \in \mathcal{T}_{X}\\ &\Longleftrightarrow\ \forall A \subset X,\ f(\bar{A}) \subset \overline{f(A)}\\ &\Longleftrightarrow\ \forall (X\setminus B) \in \mathcal{T}_{Y},\ (X\setminus f^{-1}(B)) \in \mathcal{T}_{X}\\ &\Longleftrightarrow\ \forall x \in X,\ \exists \mathcal{N}_{x}\ \text{s.t.}\ f(\mathcal{N}_{x}) \subset \mathcal{N}_{f(x)} \end{aligned}$$ where $\mathcal{N}_{x}$ is a [[Neighborhood]] of $x$. A function is continuous if and only if the inverse image of any arbitrary open set in [[Codomain]] is an open set. ## Continuity of Real-Valued Function ### Continuous at a Point Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$, and $\mathbf{x}_{0} \in \mathcal{D}$ $$\begin{aligned} \lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0}) &\Leftrightarrow \forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (|\mathbf{x}-\mathbf{x}_{0}| < \delta \Rightarrow |f(\mathbf{x})-f(\mathbf{x}_{0})| < \epsilon)\\ &\Leftrightarrow \forall\epsilon>0, \exists\delta>0\ \text{s.t}\ f(B_{\delta}(\mathbf{x})) \subset B_{\epsilon}(f(\mathbf{x}))\\ &\Leftrightarrow \forall\epsilon>0, \exists\delta>0\ \text{s.t}\ B_{\delta}(\mathbf{x}) \subset f^{-1}(B_{\epsilon}(f(\mathbf{x}))) \end{aligned}$$ $f$ is continuous at a point $\mathbf{x}_{0}$ if the [[Limit of a Function|limit]] of $f(\mathbf{x})$, as $\mathbf{x}$ approaches $\mathbf{x}_{0}$, exists and is equal to $f(\mathbf{x}_{0})$ ### Continuous on an Open Interval Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$, and $X \subset \mathcal{D}$ $$\forall \mathbf{x}_{0} \in X, \lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0})$$ A function is continuous at every point in an open interval $X$ ### Continuous Function Suppose $f: \mathcal{D} \subset \mathbb{R}^{n} \to \mathbb{R}^{m}$ $$\forall \mathbf{x}_{0} \in \mathcal{D},\ \lim\limits_{\mathbf{x} \to \mathbf{x}_{0}} f(\mathbf{x}) = f(\mathbf{x}_{0})$$ A function is continuous at every point in its domain ### Right-Continuous Suppose $f: \mathcal{D} \to \mathbb{R}$, and $a \in \mathcal{D}$ $$\forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (0 \leq x-a < \delta \Rightarrow |f(x)-f(a)| < \epsilon) \Leftrightarrow \lim\limits_{x \to a^{+}} f(x) = f(a)$$ $f$ is right-continuous at $a$ The [[Limit of a Function#right-sided-limit|right-sided limit]] of $f(x)$, as $x$ approaches $a$ from the right side, exists and is equal to $f(a)$ ### Left-Continuous Suppose $f: \mathcal{D} \to \mathbb{R}$, and $a \in \mathcal{D}$ $$\forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathcal{D}, (0 \leq a-x < \delta \Rightarrow |f(x)-f(a)| < \epsilon) \Leftrightarrow \lim\limits_{x \to a^{-}} f(x) = f(a)$$ $f$ is left-continuous at $a$ The [[Limit of a Function#left-sided-limit|left-sided limit]] of $f(x)$, as $x$ approaches $a$ from the left side, exists and is equal to $f(a)$ # Properties ## Construction of Continuous Functions Suppose $a \in \mathcal{D}$, and $f, g: \mathcal{D} \to \mathbb{R}$ is continuous at $a$ Then, the following are satisfied - $f + g$ is continuous at $a$ - $f - g$ is continuous at $a$ - $f \cdot g$ is continuous at $a$ - $g(a) \neq 0 \Rightarrow \cfrac{f}{g}$ is continuous at $a$ # Facts ![[Lipschitz Continuity#^e0a56a|^e0a56a]] > Every continuous function $f: [a, b] \to \mathbb{R}$ is [[Darboux Integral|integrable]] ^c92817 ![[Heine-Cantor Theorem]] > Given two continuous functions $f: D_{f} \subset \mathbb{R}^{n} \to R_{f} \subset D_{g}$ and $g: D_{g} \subset \mathbb{R}^{m} \to R_{g} \subset \mathbb{R}^{l}$, then their composition $h := g \circ f: D_{f} \subset \mathbb{R}^{n} \to R_{g} \subset \mathbb{R}^{l}$ is continuous. > A [[Constant Function]] $f: X \to Y,\quad f(x)=y_{0}$ is continuous. > [[Inclusion Function|Inclusion Map]] $j: A \hookrightarrow X,\quad j(x) = x$ where $A \subset X$, is continuous > $(f: X \to Y)\in C^{0}, (g: Y \to Z)\in C^{0}\ \Rightarrow\ (g\circ f: X \to Z) \in C^{0}$ > A [[Function Composition|composite function]] of continuous functions is continuous > $(f: X \to Y) \in C^{0}, A \subset X\ \Rightarrow\ (f|_{A}: A \to Y) \in C^{0}$ > A continuous function with restricted [[Domain]] is continuous > $(f: X \to Y) \in C^{0}, f(x) \subset Z \subset Y\ \Rightarrow\ (g: X \to Z,\quad g(x)=f(x)) \in C^{0}$ > $(f: X \to Y) \in C^{0}, Y \subset Z\ \Rightarrow\ (h: X \to Z,\quad h(x)=f(x)) \in C^{0}$ > A continuous function with expanded and restricted [[Codomain]] is continuous. > $(f|_{U_\alpha}: U_{\alpha}\to Y\ \text{s.t.}\ \bigcup_{\alpha} U_{\alpha} = X) \in C^{0} \Rightarrow (f: X \to Y) \in C^{0}$ > Consider a collection $(U_\alpha)$ of [[Open Set|open sets]] in $X$. If $X = \bigcup_{\alpha} U_{\alpha}= X$ and $f|_{U_\alpha}$ is continuous, then $f: X \to Y$ is continuous. ![[Gluing Lemma]] > Consider [[Topological Space|topological Spaces]] $(A, \mathcal{T}), (X \times Y, \mathcal{T})$ and a function $f: A \to X\times Y,\quad f(a) = (f_{1}(a), f_{2}(a))$. > $f \in C^{0} \Leftrightarrow (f_{1}: A \to X), (f_{2}: A \to Y) \in C^{0}$ > A function with a [[Product Topology]] [[Codomain]] is continuous if and only if all of its coordinate functions are continuous. > Consider [[Topological Space|topological Spaces]] $(A, \mathcal{T}), (\prod\limits_{i \in \mathbb{N}}X_{i}, \mathcal{T})$ and a function $f: A \to \prod\limits_{i \in \mathbb{N}}X_{i},\quad f(a) = (f_{1}(a), f_{2}(a), \dots, f_{n}(a), \dots)$. > $f \in C^{0} \Leftrightarrow \forall i \in \mathbb{N},\ (f_{i}: A \to X_{i}), \in C^{0}$ > A function with a countably infinite [[Product Topology]] [[Codomain]] is continuous if and only if all of its coordinate functions are continuous. ![[Closed Map#^b9de02|^b9de02]]$ Link to original

Discontinuity
Definition

Removable Discontinuity

Suppose $f : D \to R$ , and $a \in D$ $x \to a^{+} lim f (x) = x \to a^{-} lim f (x) \neq = f (a)$ where $a$ is called a removable discontinuity

Two one-sided limits exist and are equal, but the function value is not equal

Jump Discontinuity

Suppose $f : D \to R$ , and $a \in D$ $x \to a^{+} lim f (x) \neq = x \to a^{-} lim f (x)$ where $a$ is called a jump discontinuity

Two one-sided limits exist, but not equal

Essential Discontinuity

Suppose $f : D \to R$ , and $a \in D$ $∄ x \to a^{+} lim f (x) \lor ∄ x \to a^{-} lim f (x)$ where $a$ is called an essential discontinuity

At least one of the two one-sided limits does not exist in $R$
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Uniform Continuity
Definition

Consider a function $f : D \to R$ . The function $f$ is uniformly continuous in the $D$ if $\forall ϵ > 0, \exists δ > 0, \forall x, y \in D, (∣ x - y ∣ < δ \Rightarrow ∣ f (x) - f (y) ∣ < ϵ)$

Facts

If a function is uniformly continuous, then the function is continuous

Continuously differentiable $\subset$ Lipschitz continuous $\subset$ $α -$ Holder Continuous $\subset$ Uniformly Continuous $\subset$ Continuous Lipschitz continuous Absolute continuous $\subset$ Uniformly Continuous $\subset$ Continuous where $0 < 0 \leq 1$

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Extreme Value Theorem
Definition

Definition in Order Topology

Consider a Continuous Function $f : (X, T_{X}) \to (Y, T_{Y})$ between topological spaces where $Y$ is an ordered set in the Order Topology $T_{Y}$ . Then, if $X$ is compact, then $\exists c, d \in X s.t. \forall x \in X, f (c) \leq f (x) \leq f (d)$

Definition in Real Numbers

If $f$ is continuous on a $[a, b]$ , then $\exists a_{0}, b_{0} \in [a, b], \forall x \in [a, b], f (a_{0}) \leq f (x) \leq f (b_{0})$

If a real-valued function $f$ is continuous on the closed interval $[a, b]$ , then $f$ must attain a Extremum, each at least once.
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Intermediate Value Theorem
Definition

If $f$ is continuous in $[a, b]$ , and $f (a) < f (b)$ , then $\forall p \in (f (a), f (b)), \exists c \in (a, b), f (c) = p$

If $f$ is a Continuous Function, then it takes on any given value between $f (a)$ and $f (b)$ at some point within the interval
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Extremum
Definition

Maximum
Definition

Consider an Partially Ordered Set $X$ and a subset $Y \subset X$ . The largest element (maximum) $y_{0}$ of $Y$ is defined as $y_{0} s.t. y_{0} \in Y and \forall y \in Y, y \leq y_{0}$
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Minimum
Definition

Consider an Partially Ordered Set $X$ and a subset $Y \subset X$ . The smallest element (minimum) $y_{0}$ of $Y$ is defined as $y_{0} s.t. y_{0} \in Y and \forall y \in Y, y \geq y_{0}$
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Local Maximum
Definition

Suppose $f : D \to R$ . The function $f$ has local maximum at $x_{0}$ if $\exists ϵ > 0, (\forall x \in D, ∣ x_{0} - x ∣ < ϵ) \Rightarrow f (x) \leq f (x_{0})$
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Local Minimum
Definition

Suppose $f : D \to R$ . The function $f$ has local minimum at $x_{0}$ if $\exists ϵ > 0, (\forall x \in D, ∣ x_{0} - x ∣ < ϵ) \Rightarrow f (x_{0}) \leq f (x)$
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Derivative
Definition

Derivative

Suppose $f : (a, b) \to R$ , and $c \in (a, b)$ $f^{'} (c) = Δ x \to 0 lim \frac{Δ y}{Δ x} = x \to c lim \frac{f ( x ) - f ( c )}{x - c} = Δ \to 0 lim \frac{f ( c + Δ x ) - f ( c )}{Δ x}$

If the $f^{'} (c)$ exists, then the function $f$ is differentiable at a point $c$

Right-Derivative

Suppose $f : [a, b) \to R$ $f_{+}^{'} (a) = Δ \to 0^{+} lim \frac{f ( a + Δ x ) - f ( a )}{Δ x}$

If the $f_{+}^{'} (a)$ exists, then the function $f$ is right differentiable at a point $a$

Left-Derivative

Suppose $f : (a, b] \to R$ $f_{-}^{'} (b) = Δ \to 0^{-} lim \frac{f ( b + Δ x ) - f ( b )}{Δ x}$

If the $f_{-}^{'} (b)$ exists, then the function $f$ is left differentiable at a point $b$

Derivative Function

Suppose $f : D \to R$ , and $f$ is differentiable $\forall x \in D$ , $f^{'} (x) = \frac{df}{d x} = y \to x lim \frac{f ( y ) - f ( x )}{y - x}$

$f^{'} (x)$ is called the derivative function or the derivative of $f$

Properties

Rules of Computation

If $f, g : D \to R$ is differentiable at $a \in D$ , then $f + g, f - g, \frac{f}{g}$ also differentiable, and the following are satisfied

Linearity: $(a f \pm b g)^{'} (x) = a f^{'} (x) \pm b g^{'} (x)$

product rule: $(f g)^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)$

quotient rule: $(\frac{f}{g})^{'} (x) = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{{ g ( x ) } ^{2}}$ , where $g (x) \neq = 0$

Chain Rule: $(g \circ f)^{'} (a) = g^{'} (f (a)) f^{'} (a)$

Facts

If $f$ is differentiable at $a$ , then $f$ must also be continuous at $a$

Suppose $f : D \to R$ If $f$ has an Extremum at $a \in D$ and differentiable, then $f^{'} (a) = 0$

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Rolle's Theorem
Definition

If $f : [a, b] \to R$ is continuous and differentiable on the $(a, b)$ , then $f (a) = f (b) \Rightarrow \exists c \in (a, b), f^{'} (c) = 0$
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Mean Value Theorem
Definition

Mean Value Theorem

If $f : [a, b] \to R$ is continuous on the $[a, b]$ , and differentiable on the $(a, b)$ , then $\exists c \in (a, b), f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$

Cauchy’s Mean Value Theorem

If $f, g : [a, b] \to R$ is continuous on the $[a, b]$ , and differentiable on the $(a, b)$ , then $\exists c \in (a, b), (f (b) - f (a)) g^{'} (c) = (g (b) - g (a)) f^{'} (c)$ , or $\exists c \in (a, b), \frac{g ^{'} ( c )}{f ^{'} ( c )} = \frac{g ( b ) - g ( a )}{f ( b ) - f ( a )}$ if $f (a) \neq = f (b) \land f^{'} (c) \neq = 0$

Mean Value Theorem for Definite Integrals

If $f : [a, b] \to R$ is continuous on the $[a, b]$ , then $\exists c \in (a, b), \int_{a}^{b} f (x) d x = f (c) (b - a)$
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L'Hospital's Rule
Definition

Suppose $f, g : D \to R$ is differentiable on the $D - {a}$ , $x \to a lim f (x) = x \to a lim g (x) = 0$ or $x \to a lim ∣ f (x) ∣ = lim_{x \to a} ∣ g (x) ∣ = \infty$ , $\forall x \in D - {a}, g^{'} (x) \neq = 0$ , and $x \to a lim \frac{f ^{'} ( x )}{g ^{'} ( x )}$ exists, then $x \to a lim \frac{f ^{'} ( x )}{g ^{'} ( x )} = L \Rightarrow x \to a lim \frac{f ( x )}{g ( x )} = L$ where $L \in R \cup {- \infty, \infty}$
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Darboux Integral
Definition

Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable $\Leftrightarrow$ it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.

Darboux Sums

A partition of an interval $[a, b]$ is a finite sequence of values $x_{i}$ such that $a = x_{0} < x_{1} < \dots < x_{n} = b$ Each interval $[x_{i - 1}, x_{i}]$ is called a sub-interval of the partition.

Let $f : [a, b] \to R$ be a bounded function, $P = (x_{0}, \dots, x_{n})$ be a partition of $[a, b]$ , $Δ x_{i} = x_{i} - x_{i - 1}$ $M_{i} = sup {f (x) ∣ x_{i - 1} \leq x \leq x_{i}}$ , and $m_{i} = in f {f (x) ∣ x_{i - 1} \leq x \leq x_{i}}$

The upper Darboux sum of $f$ with respect to $P$ is $U_{f, P} = i = 1 \sum n M_{i} Δ x_{i}$

The lower Darboux sum of $f$ with respect to $P$ is $L_{f, P} = i = 1 \sum n m_{i} Δ x_{i}$

Darboux Integrals

The upper Darboux integral of $f$ is $U_{f} = \overline{\int_{a}^{b}} f = \overline{\int_{a}^{b}} f (x) d x = in f {U_{f, P}}$

The lower Darboux integral of $f$ is $L_{f} = \underline{\int_{a}^{b}} f = \underline{\int_{a}^{b}} f (x) d x = sup {L_{f, P}}$

If $U_{f} = L_{f} \Leftrightarrow \overline{\int_{a}^{b}} f = \underline{\int_{a}^{b}} f$ , then we call the common value the Darboux integral and set $\int_{a}^{b} f (x) d x = U_{f} = L_{f}$

We also say that $f$ is Darboux-integrable, simply integrable, or $f \in R [a, b]$ , where $R [a, b]$ is a set of integrable function on a $[a, b]$

Useful criterion for the integrability of $f$

$f \in R [a, b] \Leftrightarrow \forall ϵ > 0, \exists P, U_{f, P} - L_{f, P} < ϵ$

Properties

Refinement of a partition

When $P$ is a partition and, $P \subset P^{'}$ is satisfied, $P^{'}$ is a refinement of $P$

If $P^{'}$ is a refinement of $P$ , then $L_{f, P} \leq L_{f, P^{'}} \leq U_{f, P^{'}} \leq U_{f, P}$

If $P_{1}, P_{2}$ are two partitions of the same interval, then $L_{f, P_{1}} \leq U_{f, P_{2}}$

and It follows that $L_{f} \leq U_{f} \Leftrightarrow \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f$

Linearity

$\int_{a}^{b} (α f (x) + β g (x)) d x = α \int_{a}^{b} f (x) d x + β \int_{a}^{b} g (x) d x$ The Darboux Integration is a linear transformation

Additivity

$f \in R [a, b] \Leftrightarrow \forall c \in (a, b), f \in R [a, c] \land f \in R [c, b]$ , where $\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x$

Facts

Transclude of Continuous-Function#^c92817

$f, g \in R [a, b] \Rightarrow f \cdot g \in R [a, b]$ Be careful! $\int f \cdot g \neq = \int f \int g$

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Antiderivative
Definition

Suppose $f, F : D \to R$ , and the function $F$ satisfying the following, then it is called the antiderivative of $f$ $F^{'} (x) = f (x) \forall x \in D$
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Indefinite Integral
Definition

Suppose $f \in R [a, b]$ . A function $F$ satisfying the following is called an indefinite integral of $f$ $F (x) = \int_{a}^{x} f (t) d t \forall x \in [a, b]$ where $F : [a, b] \to R$
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Fundamental Theorem of Calculus
Definition

differentiating a function and integrating a function are inverses of each other apart from a constant value.

First Fundamental Theorem of Calculus

Suppose $f \in R [a, b]$ , and $F$ is the Indefinite Integral of $f$ on $[a, b]$ $F (x) = \int_{a}^{x} f (t) d t \forall x \in [a, b]$

$F$ is uniformly continuous on $[a, b]$

If $f$ is continuous on $[a, b]$ , then

$F$ is differentiable on the $[a, b]$ and $F^{'} (x) = f (x) \forall x \in [a, b]$

Second Fundamental Theorem of Calculus

Suppose $f \in R [a, b]$ , and $F : [a, b] \to R$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . $F^{'} (x) = f (x) \forall x \in [a, b]$ If $F$ is Antiderivative of $f$ , then $\int_{a}^{b} f (x) d x = F (b) - F (a)$
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Integration by Substitution
Definition

Suppose $g$ is differentiable on a $[a, b]$ , $g^{'} \in R [a, b]$ , and $f$ is continuous on a $g ([a, b])$ . Then, $\int_{a}^{b} f (g (t)) g^{'} (t) d t = \int_{g (a)}^{g (b)} f (x) d x$
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Integration by Parts
Definition

Suppose $f, g : [a, b] \to R$ is continuous on a $[a, b]$ , differentiable on a $(a, b)$ , and $f^{'}, g^{'} \in R [a, b]$ . Then, $\int_{a}^{b} f^{'} (x) g (x) d x = [f (x) g (x)]_{a}^{b} - \int_{a}^{b} f (x) g^{'} (x) d x = f (b) g (b) - f (a) g (a) - \int_{a}^{b} f (x) g^{'} (x) d x$
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Improper Integral
Definition

Open Interval

When the interval, subject to integral, is an open. $(a, b]$ or $[a, b)$

Suppose $f : (a, b] \to R$ , and $\forall c \in (a, b), f \in R [c, b]$ , then the improper integral of $f$ on a $(a, b]$ is defined as $\int_{a}^{b} f (x) d x = lim_{c \to a^{+}} \int_{c}^{b} f (x) d x$

Suppose $f : [a, b) \to R$ , and $\forall c \in (a, b), f \in R [c, b]$ , then the improper integral of $f$ on a $[a, b)$ is defined as $\int_{a}^{b} f (x) d x = lim_{c \to b^{-}} \int_{a}^{c} f (x) d x$

If there exists a limit of the expression, we call $f$ is improper integrable.

If $f : [a, b] - {c} \to R$ is improper integrable on both $[a, c)$ and $(c, b]$ , then $f$ is improper integrable on a $[a, b]$ and define $\int_{a}^{b} f (x) d x = lim_{p \to c^{-}} \int_{a}^{p} f (x) d x + lim_{q \to c^{+}} \int_{q}^{b} f (x) d x$

Unbounded Interval

When the interval, subject to integral, is an unbounded. $(- \infty, b]$ or $[a, \infty)$

Suppose $f : [a, \infty) \to R$ , and $\forall c \in R, a < c \Rightarrow f \in R [a, c]]$ , then the improper integral of $f$ on a $[a, \infty)$ is defined as $\int_{a}^{\infty} f (x) d x = lim_{c \to \infty} \int_{a}^{c} f (x) d x$

Suppose $f : (- \infty, b] \to R$ , and $\forall c \in R, c < b \Rightarrow f \in R [c, b]]$ , then the improper integral of $f$ on a $(- \infty, b]$ is $\int_{- \infty}^{b} f (x) d x = lim_{c \to - \infty} \int_{c}^{b} f (x) d x$

If there exists a limit of the expression, we call $f$ is improper integrable.

If $f$ is improper integrable on both $(- \infty, p]$ and $[p, \infty)$ , then $f$ is improper integrable on a $R$ and define $\int_{- \infty}^{\infty} f (x) d x = \int_{- \infty}^{p} f (x) d x + \int_{p}^{\infty} f (x) d x$
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Riemann–Stieltjes Integral
Definition

For a bounded function $f$ , a partition $P$ of a non-decreasing function $g$ on $[a, b]$ ,

Let $f$ be a bounded function, $P$ be a partition of a non-decreasing function $g$ on a $[a, b]$ , $Δ g_{i} = g (x_{i}) - g (x_{i - 1})$ $M_{i} = sup {f (x) ∣ x_{i - 1} \leq x \leq x_{i}}$ , and $m_{i} = in f {f (x) ∣ x_{i - 1} \leq x \leq x_{i}}$

Stieltjes Sums

The upper Stieltjes sum of $f$ with respect to $P$ is $U_{f, P, g} = i = 1 \sum n M_{i} Δ g_{i}$

The lower Stieltjes sum of $f$ with respect to $P$ is $L_{f, P, g} = i = 1 \sum n m_{i} Δ g_{i}$

Stieltjes Integrals

The upper Stieltjes integral of $f$ is $U_{f, g} = \overline{\int_{a}^{b}} f (x) d g (x) = in f {U_{f, P, g}}$

The lower Stieltjes integral of $f$ is $L_{f, g} = \underline{\int_{a}^{b}} f (x) d g (x) = sup {L_{f, P, g}}$

If $U_{f, g} = L_{f, g} \Leftrightarrow \overline{\int_{a}^{b}} f (x) d g (x) = \underline{\int_{a}^{b}} f (x) d g (x)$ , then we call the common value the Riemann–Stieltjes integral and set $\int_{a}^{b} f (x) d g (x) = U_{f, g} = L_{f, g}$

We also say that $f$ is Riemann–Stieltjes-integrable or $f \in R_{g} [a, b]$ , where $R_{g} [a, b]$ is a set of Riemann–Stieltjes integrable function on a $[a, b]$

Facts

Suppose $f \in R [a, b]$ , $g$ is non-decreasing differentiable function, $g^{'} \in R_{g} [a, b]$ , then $f \in R_{g} [a, b]$ and the following is satisfied $\int_{a}^{b} f (x) d g (x) = \int_{a}^{b} f (x) g^{'} (x) d x$

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Sequence
Definition

$(a_{n}) = f : N \to R$ $a_{m} = f (m)$

A function from natural numbers to the element at each position. The notation of a sequence can be generalized to an indexed family.

Subsequence

$(a_{n})_{n \in N} = (a_{n_{k}})_{k \in N}$ , where $(n_{k})_{k \in N}$

A sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements

Increasing and decreasing

Monotonically Increasing Sequence

$(n_{k}) \forall k \in N, n_{k} \leq n_{k + 1}$

Monotonically Decreasing Sequence

$(n_{k}) \forall k \in N, n_{k} \geq n_{k + 1}$

Strictly Monotonically Increasing Sequence

$(n_{k}) \forall k \in N, n_{k} < n_{k + 1}$

Strictly Monotonically Decreasing Sequence

$(n_{k}) \forall k \in N, n_{k} > n_{k + 1}$

Bounded

If $\exists M > 0, \forall n \in N, ∣ a_{n} ∣ \leq M$ , then the sequence $(a_{n})$ is said to be bounded.
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Limit of a Sequence
Definition

Convergence of Sequence

$lim_{n \to \infty} a_{n} = a \Leftrightarrow \forall ϵ > 0, \exists N \in N, \forall n \geq N, ∣ a_{n} - a ∣ < ϵ$

Suppose $a \in R$ , If $\forall ϵ > 0, \exists N \in N, \forall n \geq N, ∣ a_{n} - a ∣ < ϵ$ , then $(a_{n})$ is said to converge to the limit $a$

Divergence of Sequence

If $\exists ϵ > 0, \forall N \in N, \exists n \geq N, ∣ a_{n} - a ∣ \geq ϵ$ , then $(a_{n})$ is said to be divergent

Properties

Arithmetic Operations

If $n \to \infty lim a_{n}, n \to \infty lim b_{n}$ exists, then the following is satisfied

$n \to \infty lim (a_{n} \pm b_{n}) = n \to \infty lim a_{n} \pm n \to \infty lim b_{n}$

$n \to \infty lim c a_{n} = c \cdot n \to \infty lim a_{n}$

$n \to \infty lim (a_{n} \cdot b_{n}) = (n \to \infty lim a_{n}) \cdot (n \to \infty lim b_{n})$

$n \to \infty lim (\frac{a _{n}}{b _{n}}) = \frac{n \to \infty lim a _{n}}{n \to \infty lim b _{n}}$ , where $n \to \infty lim b_{n} \neq = 0$

$n \to \infty lim a_{n}^{P} = (n \to \infty lim a_{n})^{P}$

Facts

If $(a_{n})$ converges, then the limit is unique.

Squeeze Theorem for Sequences

Suppose $(a_{n}), (c_{n})$ be two sequences converging to $L \in R$ , and $(b_{n})$ be a Sequence.

If $n \to \infty lim a_{n} \leq n \to \infty lim b_{n} \leq n \to \infty lim c_{n}$ , then $n \to \infty lim b_{n} = L$
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Cauchy Sequence
Definition

A Cauchy sequence is a Sequence whose elements become arbitrarily close to each other as the sequence progresses. $(a_{n}) s.t. \forall ϵ > 0, \exists N \in N s.t. (\forall m, n \in N s.t. m, n > N, ∣ a_{m} - a_{n} ∣ < ϵ)$

Facts

Every convergent sequence is a Cauchy sequence

In $R^{n}$ space, Cauchy sequence converges.

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Completeness
Definition

A Metric Space $(X, T, d)$ is complete If and only if every Cauchy Sequence on $X$ converges to an element of $X$ .

Facts

Consider a complete Metric Space $(X, T, d)$ and a subspace $A \subset X$ . The subspace $A$ is complete if and only if $A$ is a closed set.

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Monotone Convergence Theorem
e

Definition

If a sequence is monotonically increasing and bounded above, then the sequence converges to the Supremum
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Cantor's Intersection Theorem
Definition

Cantor’s Intersection Theorem

Consider a Topological Space $(X, T)$ and a nested sequence $(E_{n})_{n \in N} s.t. \forall n \in N, E_{n + 1} \subset E_{n}$ of non-empty Compact closed subsets of $X$ . Then, $n \in N ⋂ E_{n} \neq = \emptyset$

Nested Intervals Theorem

Consider a Sequence $(I_{n})_{n \in N}$ , where $I_{n} = [a_{n}, b_{n}]$ , of closed intervals. The sequence of intervals $(I_{n})_{n \in N}$ is called a sequence of nested intervals if $\forall n \in N, I_{n + 1} \subset I_{n} and n \to \infty lim (b_{n} - a_{n}) = 0$

If $(I_{n})_{n \in N}$ is a sequence of nested intervals in real numbers, then $\exists! x \in R, x \in n \in N ⋂ I_{n}$

It is the special case of the Cantor’s intersection theorem.
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Squeeze Theorem
Definition

Squeeze Theorem for Functions

Suppose $f, g, h : D \to R$ , $L \in R$ , and $\forall x \in D - {a}, f (x) \leq g (x) \leq h (x)$

If $x \to a lim f (x) = x \to a lim h (x) = L$ , then $x \to a lim g (x) = L$

Squeeze Theorem for Sequences

Suppose $(a_{n}), (c_{n})$ be two sequences converging to $L \in R$ , and $(b_{n})$ be a Sequence.

If $n \to \infty lim a_{n} \leq n \to \infty lim b_{n} \leq n \to \infty lim c_{n}$ , then $n \to \infty lim b_{n} = L$
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Bolzano–Weierstrass Theorem
Definition

Infinite bounded Sequence in $R$ has a convergent subsequence
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Series
Definition

$n = 1 \sum \infty a_{n}$ where $a_{n}$ is a term of a Sequence $(a_{n})$

The operation of adding the terms of Sequence one after another.

Partial sum

$S_{n} = k = 1 \sum n a_{k}$

The summation of the $n$ first terms of a Sequence

Rearranged Series

$\sum_{n = 1}^{\infty} a_{f (n)}$ where $f : N \to N$ is Bijective function.

A series that is obtained by changing the order of terms in an original series’ Sequence.
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Limit of a Series
Definition

$n \to \infty lim S_{n} = S \Rightarrow n = 1 \sum \infty a_{n} = S$

A Series $n = 1 \sum \infty a_{n}$ is said to be convergent when the Sequence of partial sums $(S_{n})$ has a finite limit $S \in R$ . Otherwise, the series is said to be divergent.

Properties

Absolute Convergence

Absolute Convergence
Definition

A Series $n = 1 \sum \infty a_{n}$ converges absolutely if the series of absolute values $n = 1 \sum \infty ∣ a_{n} ∣$ converges.
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Conditional Convergence

A Series $n = 1 \sum \infty a_{n}$ is conditionally convergent if it is convergent but not absolutely convergent.

Linearity

Suppose $α, β, λ, μ \in R$ , and $n = 1 \sum \infty a_{n} = α, n = 1 \sum \infty b_{n} = β$ Then, $n = 1 \sum \infty (λ a_{n} \pm μ b_{n}) = λ α \pm μ β$

Facts

If $n = 1 \sum \infty a_{n}$ converges, then $n \to \infty lim a_{n} = 0$

Suppose $n = 1 \sum \infty a_{n}$ is a series, and $n = 1 \sum \infty a_{f (n)}$ is its rearranged series. $n = 1 \sum \infty a_{n} = n = 1 \sum \infty ∣ a_{n} ∣ = L \Rightarrow \forall f, n = 1 \sum \infty a_{f (n)} = L$

When a Series is absolutely convergent, any rearranged series will still converge to the same value.

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Sequence of Functions
Definition

Suppose $\emptyset \neq = D \subset R$ , and $\forall n \in N, f_{n} : D \to R$ , then $(f_{n})$ is called a sequence of function on $D$

Series of Functions

$n = 1 \sum \infty f_{n}$ where $f_{n}$ is a term of a sequence of function $(f_{n})$

Facts

Uniform Limit Theorem
Definition

Consider metric spaces $(X, d)$ and $(X, d)$ , and a sequence of Continuous Function $(f_{n})$ between them $f : X \to Y$ . If the sequence $(f_{n})$ uniformly converges to $f : X \to Y$ , then the function $f$ is a Continuous Function.
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Power Series
Definition

$\sum_{n = 0}^{\infty} a_{n} (x - c)^{n}$

Facts

The function $f (x) = n = 0 \sum \infty a_{n} (x - c)^{n}$ is continuous on its convergence interval.

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Analytic Function
Definition

Analytic at a point

$\forall δ > 0, \exists f (x) = n = 0 \sum \infty a_{n} (x - c)^{n}$ on a $(c - δ, c + δ)$ , then $f$ is analytic at a point $c$

If $f$ can be expressed as a power series on a $(c - δ, c + δ)$ for all $δ > 0$ , then $f$ is analytic at a point $c$

Analytic function

If $f$ is analytic at a point $x$ in any open interval $I$ , then $f$ is called an analytic function

Properties

Operations

The sums, subtractions, products, and compositions of analytic functions are analytic

Suppose $f$ is analytic on $I$ , and $g$ is analytic on $J$ $f \pm g$ , $f \cdot g$ is analytic on $I \cap J$ $\forall x_{0} \in I \cap J, g (x_{0}) \neq = 0$ , $\frac{f}{g}$ is analytic on $\forall ϵ > 0, (x_{0} - ϵ, x_{0} + ϵ)$

Suppose $f$ is analytic on $I$ , $g$ is analytic on $J$ , and $f (I) \subset J$ Then, $g \circ f$ is analytic on the $I$

Facts

A real analytic function is infinitely differentiable

Analytic function can be expressed as Taylor Series

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Pointwise Convergence
Definition

$n \to \infty lim f_{n} = f$

Suppose $f : D \to R$ , is a function, and $(f_{n})$ is a Sequence of Functions whose term has the same domain as the function $f$ If $\forall ϵ > 0, \forall x \in D, \exists N \in N, \forall n \geq N, ∣ f_{n} (x) - f (x) ∣ < ϵ$ , then $(f_{n})$ converges pointwise to $f$ on $D$ , and the function $f$ is said to be the pointwise limit function of the $(f_{n})$
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Uniform Convergence
Definition

$n \to \infty lim f_{n} = f uniformly$ Suppose $f : D \to R$ , is a function, and $(f_{n})$ is a Sequence of Functions whose term has the same domain as the function $f$ If $\forall ϵ > 0, \exists N \in N, \forall x \in D, \forall n \geq N, ∣ f_{n} (x) - f (x) ∣ < ϵ$ , then $(f_{n})$ converges uniformly to $f$ on $D$

Facts

Uniformly convergent Sequence of Functions $(f_{n})$ implies Pointwise Convergence

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Cauchy's Uniform Convergence Criterion
Definition

Suppose $f_{n} : D \to R$ If $\forall ϵ > 0, \exists N \in N, (\forall x \in D, \forall m, n \in N, m > n \geq N \Rightarrow k = n + 1 \sum m f_{k} (x) < ϵ)$ , then $n = 1 \sum \infty f_{n} (x)$ is uniformly convergent on $D$
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Weierstrass M-test
Definition

Let $(f_{n})$ be a sequence of functions $f_{n} : D \to R$ , and $(M_{n})$ be a sequence of non-negative real numbers If $\exists M_{n} > 0, \forall x \in D, \forall n \in N, (∣ f_{n} (x) ∣ \leq M_{n} \land n = 1 \sum \infty M_{n} < \infty)$ , then $n = 1 \sum \infty f_{n}$ uniformly and absolutely converges on $D$
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Root Test
Definition

Suppose $n = 1 \sum \infty a_{n}$ is a power series, and let $M := n \to \infty lim sup n ∣ a_{n} ∣$ If $M < 1$ , then $n = 1 \sum \infty a_{n}$ converges absolutely If $M > 1$ , then $n = 1 \sum \infty a_{n}$ diverges
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Radius of Convergence
Definition

Suppose $n = 0 \sum \infty a_{n} (x - c)^{n}$ is a power series, and let the radius of convergence $R := \frac{1}{n \to \infty lim sup n ∣ a _{n} ∣}$ , then $n = 0 \sum \infty a_{n} (x - c)^{n}$ converges absolutely when $∣ x - c ∣ < R$ , and diverges when $∣ x - c ∣ > R$ If $a = 0$ , it is considered $R = \infty$ , and if $a = \infty$ , it is considered $R = 0$

Facts

Let $R$ be the radius of convergence of a series $n = 0 \sum \infty a_{n} (x - c)^{n}$ The series converges uniformly on any closed interval inside a $(c - R, c + R)$ . In other words, If $\forall r, 0 < r < R$ , then the series $n = 0 \sum \infty a_{n} (x - c)^{n}$ converges uniformly on a $[c - r, c + r]$

If the radius of convergence of a series $n = 0 \sum \infty a_{n} (x - c)^{n}$ is an $R$ , then the radius of convergence of $n = 1 \sum \infty n a_{n} (x - c)^{n - 1}$ will also be $R$

If the radius of convergence of a series $n = 0 \sum \infty a_{n} (x - c)^{n}$ is an $R$ , then the radius of convergence of $n = 0 \sum \infty \frac{a _{n}}{n + 1} (x - c)^{n + 1}$ will also be $R$

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Uniform Limit Theorem
Definition

Consider metric spaces $(X, d)$ and $(X, d)$ , and a sequence of Continuous Function $(f_{n})$ between them $f : X \to Y$ . If the sequence $(f_{n})$ uniformly converges to $f : X \to Y$ , then the function $f$ is a Continuous Function.
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Abel's Summation Formula
Definition

Abel’s Summation Formula for Natural Numbers

$k = m \sum n a_{k} b_{k} = a_{n} k = m \sum n b_{k} + j = m \sum n - 1 ((a_{j} - a_{j + 1}) k = m \sum j b_{k})$
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Abel's Limit Theorem
Definition

Suppose a Power Series $n = 0 \sum \infty a_{n} (x - c)^{n}$ has the Radius of Convergence $R$ . If the series converges at the boundary points $c \pm R$ , then it converges uniformly on any closed interval inside a $[c - R, c + R]$
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Differentiation of Sequence of Functions
Definition

Differentiation of Sequence of Functions

Let $f := n \to \infty lim f_{n}$

If $(f_{n})$ is a sequence of differentiable function on a bounded interval $I$ ,

$\forall x_{0} \in I, \exists n \to \infty lim f_{n} (x_{0}) \land (∣ f (x_{0}) ∣ < \infty)$ , and

$(f_{n}^{'})$ converges uniformly on the $I$ ,

Then $(f_{n})$ converges uniformly on the interval $I$ , the limit function $f$ is differentiable on $I$ , and $\forall x_{0} \in I, f^{'} (x_{0}) = n \to \infty lim f_{n}^{'} (x_{0})$

Differentiation of Series of Functions

Let $f := n = 1 \sum \infty f_{n}$

If $(f_{n})$ is a sequence of differentiable function on a bounded interval $I$ ,

$\forall x_{0} \in I, \exists n = 1 \sum \infty f_{n} (x_{0}) \land (∣ f (x_{0}) ∣ < \infty)$ , and

$n = 1 \sum \infty f_{n}^{'}$ converges uniformly on the $I$ ,

Then $f$ converges uniformly, and differentiable on the interval $I$ , and $\forall x_{0} \in I, f^{'} (x_{0}) = n = 1 \sum \infty f_{n}^{'} (x_{0})$
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Differentiation of Power Series
Definition

Suppose a power series $f (x) = n = 0 \sum \infty a_{n} (x - c)^{n}$ has a Radius of Convergence $R$ Then, $f$ is differentiable on a $(c - R, c + R)$ , and the derivative of $f$ is $f^{'} (x) = n = 1 \sum \infty n a_{n} (x - c)^{n - 1}$
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Integration of Sequence of Functions
Definition

Integration of Sequence of Functions

If $\forall n \in N, f_{n} \in R [a, b]$ , and $(f_{n})$ converges uniformly to $f$ on an interval $[a, b]$ , then $f \in R [a, b]$ , and $n \to \infty lim \int_{a}^{b} f_{n} (x) d x = \int_{a}^{b} f (x) d x$ where $R [a, b]$ is a set of integrable function on a $[a, b]$

Term by Term Integration

If $\forall n \in N, f_{n} \in R [a, b]$ , and $n = 1 \sum \infty f_{n}$ converges uniformly to $f$ on $[a, b]$ , then $f \in R [a, b]$ , and $\int_{a}^{b} f (x) d x = n = 1 \sum \infty \int_{a}^{b} f_{n} (x) d x$ where $R [a, b]$ is a set of integrable function on a $[a, b]$
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Integration of Power Series
Definition

If a function $f (x) = n = 0 \sum \infty a_{n} (x - c)^{n}$ converges on $[a, b]$ , then $f \in R [a, b]$ , and $\int_{a}^{b} f (x) d x = n = 0 \sum \infty a_{n} \int_{a}^{b} (x - c)^{n} d x$ where $R [a, b]$ is a set of integrable function on a $[a, b]$

Improper Integral of Power Series

If a function $f (x) = n = 0 \sum \infty a_{n} (x - c)^{n}$ , and the power series $n = 0 \sum \infty \frac{a _{n}}{n + 1} (b - c)^{n + 1}$ converges on $[a, b)$ , then $f$ is improper integrable on the $[a, b)$ , and $\int_{a}^{b} f (x) d x = n = 0 \sum \infty a_{n} \int_{a}^{b} (x - c)^{n} d x$
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Problems

Limit of a function
pointwise continuous
Uniform continuity
Extreme value theorem
Intermediate value theorem
local minimum and maximum
Rolle’s theorem
Mean value theorem
Cauchy mean value theorem
L’Hopital’s Rule
Integral function’s conditions
Antiderivative
Indefinite integral
Fundamental theorem of calculus
- 1st
- 2nd
Convergence of sequence
Cauchy’s sequence
Monotone convergence theorem
Nested interval theorem
Bolzano-Weierstrass theorem
Absolute, conditional convergence
Rearranged array’s converged value
Definition of analytic function
Pointwise convergence
Uniformly convergence
Cauchy’s uniform convergence criterion
Weierstrass M-test
Root test
Radius of convergence, convergence interval
Uniform limit theorem
Abel’s limit theorem
Conditions of differentiation of function series
Differentiation of power series
Conditions of integration of function series
Term by Term integration
Integration of power series

My Knowledge Base

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Mathematical Analysis Note

Archimedean Property

Definition

Proof

Dense Set

Definition

Denseness of the Rational Numbers

Denseness of the Irrational Numbers

Real Number

Definition

Construction

Facts

Law of Trichotomy

Definition

Field

Definition

Order Axiom

Definition

Completeness Axioms

Definition

Facts

Construction of the Real Numbers

Definition

Axiomatic Definition

Construction from Cauchy Sequences

Construction by Dedekind Cuts

Operations

Limit of a Function

Definition

Function of a Single Variable

Epsilon-Delta-Definition of Limit

Right-Sided Limit

Left-Sided Limit

Limits at infinity

Infinite Limits

Function of Multi Variables

Ordinary Limits

Properties

Algebraic Limit Theorem

Facts

Squeeze Theorem for Functions

Continuous Function

Definition

Topological Definition of Continuity

Discontinuity

Definition

Removable Discontinuity

Jump Discontinuity

Essential Discontinuity

Uniform Continuity

Definition

Facts

Extreme Value Theorem

Definition

Definition in Order Topology

Definition in Real Numbers

Intermediate Value Theorem

Definition

Extremum

Definition

Maximum

Definition

Minimum

Definition

Local Maximum

Definition

Local Minimum

Definition

Derivative

Definition

Derivative

Right-Derivative

Left-Derivative

Derivative Function

Properties

Rules of Computation

Facts

Rolle's Theorem