Definition:Piecewise Continuous Function

Definition
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Then $f$ is piecewise continuous if:
 * $(1): \quad$ there exists a finite partition $\left\{{x_0, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, $x_0 = a$ and $x_n = b$, such that $f$ is continuous on $\left({x_{i−1} \,.\,.\, x_i}\right)$ for every $i \in \left\{{1, \ldots, n}\right\}$
 * $(2): \quad$ the one-sided limits $\displaystyle \lim_{x \to x_{i−1^+}} f\left({x}\right)$ and $\displaystyle \lim_{x \to x_{i^-}} f\left({x}\right)$ exist for every $i \in \left\{{1, \ldots, n}\right\}$.

Properties

 * A piecewise continuous function is bounded. (Stated in Myint-U and Debnaths' book.)
 * A piecewise continuous function is integrable. (Stated in Myint-U and Debnaths' book.)

Comments
There are other definitions of Piecewise Continuous Function. For example, the following variations exist:
 * $f$ need not be defined at the points $x_i$.
 * The partition in (1) above can be infinite when the domain of $f$ is unbounded.
 * The requirement in (2) above is replaced by the requirement that $f$ be bounded.
 * The codomain of $f$ is $\C$ instead of $\R$.

Possible properties of piecewise continuous functions:


 * It seems obvious that a linear combination, a product, a quotient, and a composite of piecewise continuous functions are piecewise continuous functions.
 * The Extreme Value Theorem should hold for piecewise continuous functions.
 * A piecewise continuous function should be piecewise uniformly continuous.