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Let $a<c<b$, $f:(a,b)\to \mathbb R$ be continuous. Assume that $f$ is differentiable at every point of $(a,b)\setminus\{c\}$ and $f'$ has a limit at $c$. Then Which of the following are correct?

(A) $f$ is differentiable at $c$.

(B) $f$ need not be differentiable at $c$

(C) $f$ is differentiable at $c$ and $\lim_{x\to c}f'(x)=f'(c)$

(D) $f$ is differentiable at $c$ but $f'(c)$ is not necessorily $\lim_{x\to c}f'(x)$

Here, $\lim_{x\to c^+}f'(x)=\lim_{x\to c^-}f'(x)=L$(given). I tried to draw a curve satisfying this with sharp points(My aim is to show that $f$ need not be differentiable). But not able to find. So, $f$ is differentiable at $c$. But I am not able to prove analytically.

Math geek
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1 Answers1

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By the mean value theorem, there is $k_c\in (a,c)$ s.t. $$\frac{f(c)-f(a)}{c-a}=f'(k_c).$$

If $L:=\lim_{x\to c}f'(x)$, then $$f'_{\text{left}}(c)=\lim_{x\to c^-}\frac{f(c)-f(a)}{c-a}=L.$$

Do as well on $(c,b)$ and you'll get that $$f_{\text{right}}'(c)=f'_{\text{left}}(c)=L.$$

Surb
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