Definition:Piecewise Continuous Function/Also defined as

Piecewise Continuous Function: Also defined as

There are other definitions of piecewise continuous function.

For example, the following variants exist:

Variant 1: $f$ is piecewise defined

Let $\closedint a b$ be a closed interval.

Let $\set {x_0, x_1, \ldots, x_n}$ be a finite subdivision of $\closedint a b$, where $x_0 = a$ and $x_n = b$.

Let $f$ be a real function defined on $\closedint a b \setminus \set {x_0, x_1, \ldots, x_n}$.

$f$ is piecewise continuous with one-sided limits if and only if:

for all $i \in \set {1, 2, \ldots, n}$:

$(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$

$(2): \quad$ the one-sided limits $\ds \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\ds \lim_{x \mathop \to {x_i}^-} \map f x$ exist.

Variant 2: $f$ is complex-valued

Let $f$ be a complex-valued function defined on a closed interval $\closedint a b$.

$f$ is piecewise continuous if and only if:

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 $\openint {x_{i − 1} } {x_i}$.

Variant 3: $f$ has unbounded domain

Let $f$ be a real function defined on $\R$.

$f$ is piecewise continuous if and only if:

for any closed interval $\closedint a b$:

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 $\openint {x_{i − 1} } {x_i}$.