# functions on intervals with endpoints

Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or just on $(0,\infty)$?

More generally, is there a standard interpretation of the assertion "$F$ is an antiderivative of $f$ on $I$" when $I$ contains one or both of its endpoints?

It could be taken to mean that $F$ is continuous on $I$ and $F'(x)=f(x)$ for all $x$ in the interior of $I$, or it could include a condition on the one-sided derivatives at the endpoint(s), specifically that the right-sided (resp. left-sided) derivative of $F$ at the left (resp. right) endpoint of $I$ should agree with the value of $f$ there.

I suspect one can find an example (of the $x^2 \sin 1/x$ variety) that would show that the two definitions are inequivalent.

This is related to the issue of what one means when one says that a function is continuous on an interval. Does one want to say that the right-continuous Heaviside function is continuous on $[0,\infty)$ or does one want to say that it is merely continuous on $(0,\infty)$? That is, should one use a one-sided limit or a two-sided limit at the endpoint? Clearly the former choice makes more sense if the function isn't even defined on the other side of the endpoint (as in the case of $\sqrt{x}$), but if the function is defined on the other side of the endpoint, then arguments could be made for the latter choice.

My guess is that these questions occupy a fuzzy zone where different authors make different choices of terminology, and where there's no consensus. But I'd be glad to learn otherwise.

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As a remark: this is why in most books/paper you only see continuity and differentiability specified on open domains with possibly the additional assumption of "... extends continuously to the boundary". Note that in more than 1 dimensions, continuous extension to the boundary is more than just issue of blow-up on the boundary point, but also continuity on the boundary which is now more than 0 dimensional. – Willie Wong Feb 19 '11 at 9:25
I think it's purely a matter of choice, with no universal agreement; in any specific example, just make sure you say "one-sided derivative" or "extendable function" etc. to make clear what you mean. I've seen a variety of different conventions in use. – Zen Harper Feb 22 '11 at 9:25