I was wondering if a function can be differentiable at its endpoint. Why so many theorems require a function to be continuous on [a,b] but only state differentiable on (a,b) not that differentiability on [a,b]

I want a generally speaking. Thank you.

Question

I was wondering if a function can be differentiable at its endpoint.  Why so many theorems require a function to be continuous on [a,b] but only state differentiable on (a,b) not that differentiability on [a,b]

I want a generally speaking. Thank you.

phi · Answer

doesn't the definition of differentiable use the idea of limits?
and a limit has to work from "both sides"
if you try to include the actual endpoint, you don't have "both sides", just the interior side.

anonymous · Answer

I get your point. But there is something called right or left limit. Is there something called right or left differentiabilit?

ganeshie8 · Answer

yes, if the function is continuous on [a, b] and differentiable on (a, b), the right derivative exists at `a` and the left derivative exists at `b` see http://mathcentral.uregina.ca/QQ/database/QQ.09.09/h/dave4.html

ganeshie8 · Answer

The theorems that don't require the differentiability at endpoints use open interval. Obviously they work on closed intervals also. They just don't care whether the function is differentaible at endpoints or not.

anonymous · Answer

So you mean that if a question state a function which is differentiable on (a,b), we can not know whether the right derivative or left derivative exists on the endpoints?