TITLE: What's the difference between continuous and piecewise continuous functions?
QUESTION [5 upvotes]: A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit. 
I was looking at the image of a piecewise continuous function on the following page: http://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx 
But the image of the function they've presented isn't continuous. As such, I'm confused by what a piecewise continuous function is and the difference between it and a normal continuous function.
I'd appreciate it if someone could explain the difference between a continuous function and a piecewise continuous function. Also, please reference the image of the piecewise continuous function presented on this page http://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx .
Thank you.

REPLY [2 votes]: $\newcommand{\R}{\mathbb{R}}$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.