Continuity of derivative implies differentiability at a point

Question

Suppose $I\subset\Bbb{R}$ is an open interval, $a\in I$ and $f:I\to\Bbb{R}$ is continuous. Suppose also that $f$ is diffble on $I-\{a\}$. Show that if $\lim_{x\to a} f'(x)=s$ exists, $f$ is diffble at $a$ and $f'(a)=s$.

I honestly don't know where to start. Any hints?

Martín-Blas Pérez Pinilla · Answer 1 · 2015-03-26T07:15:39.730

1

Hint: $f$ is

continuous in $[a,a+\epsilon]$ and $[a−\epsilon,a]$,

differentiable in $(a,a+\epsilon)$ and $(a-\epsilon,a)$;

apply the mean value theorem to estimate $$\frac{f(x)-f(a)}{x-a}.$$

edited Mar 26 '15 at 07:15

answered Mar 26 '15 at 07:08

Martín-Blas Pérez Pinilla

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$f$ is differentiable in $(a,a+\epsilon)$ and $(a-\epsilon,a)$ – Martín-Blas Pérez Pinilla Mar 26 '15 at 07:12
I don't get it. But I applied L'hospital's rule and it seems to work – sun Mar 26 '15 at 07:12

Continuity of derivative implies differentiability at a point

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