In books like Calculus (Larson), in the theorems’definitions like Rolle’s theorem, when they talk about continuity, they use closed intervals [a,b]. But when they talk about differentiability they use open brackets (a,b).

Why are closed intervals used for continuity and open intervals for differentiability?

Why can’t you say “differentiable on the closed interval [a,b]”?

Rolle’s Theorem definition

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Take for example f:[0,1]→Rf\colon[0,1]\to\mathbb R, x↦√xx\mapsto\sqrt x.

– Michael Hoppe

Oct 20 at 20:37

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2 Answers

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It is not that “closed intervals are used for continuity and open intervals for differentiability” (more on this one later). It is that, for Rolle’s Theorem (and the Mean Value Theorem), we need those hypotheses.

In the proof, we use that a continuous function on [a,b][a,b] attains a maximum. And we only need differentiability inside, so we do not need to make further assumptions on the boundary about differentiability (again, more on this later). And it is a nice exercise to see that if you relax any hypothesis on Rolle’s Theorem you do not have a true general statement anymore.

Now, continuity can be talked about in far more general settings. More particularly, we can talk about continuity on any subset of the real numbers in a rather canonical fashion (no need to be intervals, closed or open or whatever).

Differentiability is a little trickier. It is common to define differentiability only on open sets when we are in Euclidean space (not only open intervals, but open sets in general). This is partly due to the fact that being able to differentiate from every direction is a must in some theorems and some basic facts which we would like to have. However, there are cases for which talking about differentiability, in some sense, on “not-open” sets is useful and/or a must. This is true for example when talking about functions on the closed half-space (which enhances its discussion on manifolds with boundaries), or when talking about closed submanifolds of some manifold.

In your particular setting, we can define differentiability on [a,b][a,b] on many ways. Firstly, we can simply extend to the fact that the limit which defines the derivative exists on the boundaries (however, it will be only a one-sided limit). Or we can extend by saying that ff is differentiable on [a,b][a,b] if there exists a differentiable function gg on an open set containing [a,b][a,b] such that g|[a,b]=fg|_{[a,b]}=f. Instead of discussing this further, I’ll just say that differentiability is more subtle than continuity with respect to its domains.

What does g|[a,b]=fg|_{[a,b]}=f stand for? Would you please elaborate more on “And it is a nice exercise to see that if you relax any hypothesis on Rolle’s Theorem you do not have a true general statement anymore” and write an example, if possible in R2 or R3?

– ron21

Oct 20 at 19:03

In the particular case of Rolle’s theorem you need continuity on [a,b][a,b], but you only need differentiability in (a,b)(a,b). This being said, in R{\Bbb R} there is no problem in defining differentiability on [a,b][a,b] (differentiability from the right/left).

In higher dimensions this gets more complicated. It is ‘easier’ to define differentiability on an open domain. Whereas continuity may be defined without problems on the boundary of a domain. For example, think of the continuous image of a compact set being compact is an extremely useful property and really needs the definition on the compact set.