Re: f continuous on [a,b] and differentiable on (a,b)


deniz.bahar@xxxxxxxxx wrote:
> Why do many theorems in calculus/analysis have the hypotheses, f
> continuous on [a,b] and differentiable on (a,b)?
>
> My question is really why the endpoints are not included for the
> differentiable hypothesis. Aren't functions that are continous on
> [a,b] and differentiable on (a,b) also differentiable from one side at
> the endpoints?

Think of some examples and various scenarios. For example, consider the
function f(x) = sqrt(1 - x^2). For any x within [-1, 1] this is
defined, and continuous. It is differentiable in
(-1, 1), which is *open*, because...

<drumroll>

....if you take the derivative from the right at x = -1, you see that
your limit goes to -oo and thus the derivative doesn't exist. Same with
the derivative from the left at x = 1.

.

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