I thought this was only an issue in K-12, as anyone in research math considers the domain and the target to be part of a function, and then the problem disappears: The function R \ {0} -> R, x |-> 1/x is not just piecewise continuous but continuous on-the-nose. The function R -> R, x |-> 1/x doesn't exist. The function R |-> some completion of R, x |-> 1/x is continuous or not depending on which completion you choose (the one with two infinities or the projective line).
But I do recall a similar confusion happening with "piecewise linear" (the question is whether the pieces have to fit together).
> I thought this was only an issue in K-12, as anyone in research math considers the domain and the target to be part of a function
Anyone is research math does consider the domain and target to be part of the function. Alas, that doesn't mean that they'll actually write down which domain and target they have in mind.
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.
But I do recall a similar confusion happening with "piecewise linear" (the question is whether the pieces have to fit together).