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

A piecewise continuous function doesn't have to be continuous at finitely many points in a finite interval, so long as you can split the function into subintervals such that each interval is continuous.

A nice piecewise continuous function is the floor function:

The function itself is not continuous, but each little segment is in itself continuous.

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.