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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 disappea

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).

nicoburns 6y ago
> 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.
tgb 6y ago
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." Emphasis added. https://mathworld.wolfram.com/PiecewiseContinuous.html https://mathworld.wolfram.com/PiecewiseContinuous.html
- generationP 6y ago
  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?
  - tgb 6y ago
    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.

nicoburns 6y ago

> 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.

tgb 6y ago

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." Emphasis added. https://mathworld.wolfram.com/PiecewiseContinuous.html https://mathworld.wolfram.com/PiecewiseContinuous.html

generationP 6y ago
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?
- tgb 6y ago
  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.