Piecewise function

A function $f(t)$ is piecewise continuous if for an interval $I$, then $\forall t\in I$, it has a finite number of discontinuities $t_1,\cdots ,t_N$, and the limits ${\lim }_{t\to t_i^{-}}f(t)$ and ${\lim }_{t\to t_i^{+}}$ exist and are finite.

Intuitively, a piecewise function where all points are defined, with no removable continuities, are piecewise continuous. A function with a pole is not.