$\newcommand{\R}{\mathbb{R}}$The notion of piecewise continuity (PWC) is used differently in different contexts. a Often, a function $f:\R\to\R$ is called PWC if it continuous everywhere, but at a finite number of points.

In the context of Laplace transform and other integral transforms, a function $f$ is said to be PWC if it is continuous on a partition of intervals of its domain and at the boundaries of the intervals the function has well-defined and finite limits.

Definition. [PWC] A function $f:[a,b]\to\R$ is called piecewise continuous (PWC) if there exist $a = x_0 < x_1 < \ldots < x_n = b$ so that

$f$ is continuous on $(x_k, x_{k+1})$ for all $k=0,\ldots, n-1$

The limits $\lim_{x\to{}x_{k+1}^{-}}f(x)$ and $\lim_{x\to{}x_{k}^{+}}f(x)$ exist and are finite for all $k=0,\ldots, n-1$

According to this definition, function $$ f(x) = \begin{cases} 0, &\text{ for } x = 0 \\ \frac{1}{x}, &\text{ for } x{}>{}0 \end{cases} $$ defined over $[0, \infty)$, is not PWC according to the second definition although it has only one point of discontinuity.

Additionally, function $f(x)=\tfrac{1}{x}$, $x\in\R\setminus\{0\}$, is not PWC, again because the limits $\lim_{x\to 0^+}f(x)$ and $\lim_{x\to 0^-}f(x)$ are not finite.