Definition:Piecewise Continuous Function

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

$f$ is piecewise continuous :


 * there exists a finite subdivision $\left\{ {x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that:


 * for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$.

Further conditions can be placed on this definition to add specialisation as necessary:

Also defined as
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 subdivision above can be infinite when the domain of $f$ is unbounded.


 * The codomain of $f$ is $\C$ instead of $\R$.

Also see

 * Piecewise Continuous Function with One-Sided Limits is Bounded
 * Bounded Piecewise Continuous Function may not have One-Sided Limits


 * Bounded Piecewise Continuous Function has Improper Integrals
 * Piecewise Continuous Function with Improper Integrals may not be Bounded


 * Definition 3 implies Definition 4; the converse is not true

Comments
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.


 * Also, a function piecewise continuous on adjacent intervals should be piecewise continuous on the union of these intervals.