No you've missed the distinction. In all cases the domain and range are R (you can fill in a value at 0, it doesn't matter which). See the MathWorld page which leaves the definition intentionally ambiguous:
"A function or curve is piecewise continuous if it is continuous on all but a finite number of points at which certain matching conditions are sometimes required."
I don't get this. If you require f to be piecewise continuous outside of a finite set of points and to satisfy left limit = right limit at each of these points, then you just have a continuous function. Why another word for it?
The left and right limits are required to exist (and finite), but not necessarily to be equal to each other. So f(x) = 1/x and sin(1/x) are out but x/abs(x) is not.
"A function or curve is piecewise continuous if it is continuous on all but a finite number of points at which certain matching conditions are sometimes required."
Emphasis added.
https://mathworld.wolfram.com/PiecewiseContinuous.html