What's the difference between continuous and piecewise continuous functions? 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.

A function $f$ is piecewise continuous on an interval $J\subset{\mathbb R}$ if it is continuous apart from a set of isolated points $\xi\in J$ where only the one-sided limits $\lim_{x\to\xi-}f(x)$ and $\lim_{x\to\xi+} f(x)$ exist.
Note that $f(x):=\sin{1\over x}$ $(x\ne0)$ together with $f(0):=0$ does not define a piecewise continuous function on ${\mathbb R}$, even though this $f$ is continuous in the "segments" created by the special point.
