Math CentralQuandaries & Queries


Question from Dave, a student:


I was wondering if a function can be differentiable at its endpoint. For example if I have Y = X^2 and it is bounded on closed interval [1,4], then is the derivative of the function differentiable on the closed interval [1,4] or open interval (1,4). They always say in many theorems that function is continuous on closed interval [a,b] and differentiable on open interval (a,b) and an example of this is Rolle's theorem. Thank you for your help.

OK, sit down, this is complicated. A function defined (naturally or artificially) on an interval [a,b] or [a,infinity) cannot be differentiable at a because that requires a limit to exist at a which requires the function to be defined on an open interval about a.

But the relevant quotient may have a one-sided limit at a, and hence a one-sided derivative.

Moreover, we say that a function is differentiable on [a,b] when it is differentiable on (a,b), differentiable from the right at a, and differentiable from the left at b.

Experience has shown that these are the right definitions, even though they have some paradoxical repercussions. For instance, a function may be differentiable on [a,b] but not at a; and a function may be differentiable on [a,b] and [b,c] but not on [a,c].

The reason that so many theorems require a function to be continuous on [a,b] and differentiable on (a,b) is not that differentiability on [a,b] is undefined or problematic; it is that they do not need differentiability in any sense at the endpoints, and by using this looser phrasing the theorem becomes more generally applicable. So for instance you can use Rolle's theorem for the square root function on [0,1]. There are other theorems that need the stronger condition.

I would suggest, however, that whenever there is any question of a fiddly detail like this you first make sure you have the notation right and also use a few extra words to ensure the reader understands too!

Good Hunting!

About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS