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 $\R$.

Properties of Definition 1

 * Piecewise Continuous Function/Definition 1 is Bounded


 * Piecewise Continuous Function/Definition 1 is Riemann Integrable


 * Piecewise Continuous Function/Definition 1 is Uniformly Continuous on Each Piece

Properties of Definition 2

 * Piecewise Continuous Function/Definition 2 is Riemann Integrable

Relations between Definitions

 * Definition 1 implies Definition 2; the converse is not true


 * Definition 2 implies Definition 3; the converse is not true


 * 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.