Definition:Piecewise Continuous Function

Definition
Let $f$ be a real function defined on a closed interval $\closedint a b$.

$f$ is piecewise continuous :


 * there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that:


 * for all $i \in \set {1, 2, \ldots, n}$, $f$ is continuous on $\openint {x_{i − 1} } {x_i}$.

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

Also known as
Some sources hyphenate: piecewise-continuous.

Also defined as
There are other definitions of Piecewise Continuous Function.

For example, the following variants exist:

Also see

 * Bounded Piecewise Continuous Function has Improper Integrals
 * Piecewise Continuous Function with One-Sided Limits is Bounded


 * Piecewise Continuous Function does not necessarily have Improper Integrals
 * Piecewise Continuous Function with Improper Integrals may not be Bounded
 * Bounded Piecewise Continuous Function may not have One-Sided Limits

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.