In a piecewise function definition like

$$ f(x) = \begin{cases} \frac{1}{x+2} &\quad x<0 \\ 3 &\quad 0\le x\le 1 \\ 4-x^2 &\quad x>1 \end{cases} $$

what do you call the numbers where the "pieces" are "joined", i.e. the endpoints of the intervals on which each piece is defined (here, $0$ and $1$)? I can make up something of course ('break points'? 'join points'? 'joints'?), but I would like to know if there's a common term. I have looked in several books and on Wikipedia but not found any name given to these points.

Note that the function might still be continuous (or even differentiable) at these points, and might fail to be continuous or differentiable at points other than these (e.g. if one of the "pieces" already has a singularity), so one can't call them anything like "points of discontinuity".

Note also that the identification of these points is a property of the function definition rather than of the function itself (i.e. it is an "intensional" property rather than an "extensional" one): we can give a piecewise definition of a function that also has a non-piecewise definition.


marked as duplicate by smcc, Lord Shark the Unknown, Leucippus, Brahadeesh, spaceisdarkgreen Nov 17 '18 at 7:19

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  • $\begingroup$ I don't know of a standard usage, and I suggest "knots", borrowed from the definition of B-splines. $\endgroup$ – Yves Daoust Nov 16 '18 at 19:32