Definition:Piecewise Continuous Function

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

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 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 $\R$.