Definition:Piecewise Continuous Function/Variant 3

Definition

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 $\openint {x_{i − 1} } {x_i}$.