Piecewise Continuous

Definition

Function $f$ is piecewise continuous if both $f(x_{k-1}^{+})$ and $f(x_k^{-})$ where $(k=1,2,\dots ,n)$, are continuous

Notations

$C_p(a,b)$: The set^[vector space] of all piecewise continuous functions on $(a,b)$

Facts

A continuous function may not be piecewise continuous

If $f$ is piecewise continuous on $(a,b)$, we can define ${\int }_a^{b}f(x)dx={\sum }_{k=1}^n{\int }_{k-1}^{k}f(x)dx$

For any two piecewise functions $f,g$. $f+g$ is piecewise continuous and $c\cdot f$ is piecewise continuous

If $f$ is piecewise continuous on $(a,b)$, then $f$ is bounded ($\forall x\in (a,b):|f(X)|\le M$) and ${\int }_a^{b}f(x)dx$, ${\int }_a^{b}f(x)\cos lxdx$, ${\int }_a^{b}f(x)\sin lxdx$ are finite, where $l\in \mathbb{R}$