Differentiability

Definition

One dimension

A function $f:U\subset \mathbb{R}\to \mathbb{R}$ is differentiable at a point $a$ if there exists a Derivative $f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$

Higher dimensions

A function $\mathbf{f}:D\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ is differentiable at a point $\mathbf{p}\in D$ if there exists a linear map $\nabla f(\mathbf{p})$ such that ${\lim }_{||\mathbf{t}||\to 0}\frac{||f(\mathbf{p}+\mathbf{t})-f(\mathbf{p})-\nabla f(\mathbf{p})\cdot \mathbf{t}||}{||\mathbf{t}||}=0$ where $\nabla f(\mathbf{p}):{\mathbb{R}}^m\to {\mathbb{R}}^n=\left(\begin{array}{c}\nabla f_1(p)\\ ⋮\\ \nabla f_m(p)\end{array}\right)=\left(\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \dots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \dots & \frac{\partial f_m}{\partial x_n}\end{array}\right)$ is the Gradient or Jacobian Matrix

Facts

Two-Path Test for Nonexistence of a Limit

If a function $f:\mathcal{D}\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ has different limits along two different paths in the $\mathcal{D}$ as $\mathbf{x}\in \mathcal{D}$ approaches $\mathbf{a}\in \mathcal{D}$, then $\nexists \underset{\mathbf{x}\to \mathbf{a}}{\lim }f(\mathbf{x})$.