Re: Another question about derivatives

On Mon, 26 Dec 2005 15:29:32 -0800, The World Wide Wade
<waderameyxiii@xxxxxxxxxxxxxxxxxxxx> wrote:

>In article <u1m0r1tj1h80gcts8nu7m421on57rs8r3k@xxxxxxx>,
> quasi <quasi@xxxxxxxx> wrote:
>
>> On Mon, 26 Dec 2005 12:02:55 -0800, The World Wide Wade
>> <waderameyxiii@xxxxxxxxxxxxxxxxxxxx> wrote:
>>
>> >In article
>> ><1135625922.654371.258210@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>> > "Stephen J. Herschkorn" <sjherschko@xxxxxxxxxxxx> wrote:
>> >
>> >> Let f be a continuous, real-valued function on the open interval
>> >> (-1, 1). Suppose f is differentiable on (-1, 0) U (0, 1) and that
>> >> f'(x) approaches 0 as x approaches 0. Is f necessarily
>> >> differentiable at 0?
>> >
>> >Yes, by the mean value theorem.
>>
>> I don't see it.
>>
>> Can you show more of the details?
>
>[f(x) - f(0)]/(x-0) = f'(c_x) by the MVT. As x -> 0, c_x -> 0,
>which by hypothesis implies f'(c_x) -> 0.

But you want to prove f'(0) exists.

You already know that lim f'(x) --> 0 as x --> 0. That's part of the
hypothesis.

Thus, establishing that f'(c_x) --> 0 seems unproductive.

quasi
.

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