Re: Ullrich the Mathematician!
From: Mike Oliver (mike_lists_at_verizon.net)
Date: 03/08/05
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Date: Tue, 08 Mar 2005 16:33:21 -0600
N. Silver wrote:
> Jason wrote:
>>First you say "suppose f is differentiable on [0,1]" and in parentheses
>>you write "or differentiable on (0,1) and continuous on [0,1]".
>>May I remind you the two are not the *equivalent*.
>
>
> I think they are equivalent. To be differentiable at the endpoints means
> one-sided differentiability and continuity there is enough.
Continuous differentiability on an open interval, together with
continuity at the endpoints, does not buy you even one-sided
differentiability at the endpoints. You can see this from
a minor variation of the last example I posted. Let
| x sin(1/x) , 0 < x <= 1
f(x) = |
| 0, , x = 0
f is continuous at 0 and (continuously) differentiable on (0,1), but not
differentiable at 0.
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