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

1)$f$ is differentiable at $c$.

2)$f$ need not be differentiable at $c$.

3)$f$ is differentiable at $c$ but $f'(c)$ is not necessarily $\lim_{x \to c}f'(x)$.

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

Since $f$ is given to be continuous everywhere and is differentiable at every point except at $c$, I feel option 1) is correct, hence 2) is false.

How to look at 3) and 4)?

Naman
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  • With "$f'$ has a limit point at $c$", you mean that $\lim_{x\to c}f'(x)$ exists? – Hagen von Eitzen Jun 16 '19 at 16:09
  • but....can the limit $\lim_{x \to c} f'(x)$ be different from $f'(c)$? – Naman Jun 16 '19 at 16:14
  • @Hagen von Eitzen yes....I corrected the same – Naman Jun 16 '19 at 16:15
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    @Naman: The answers to the question that I linked to show that (4) holds. – Also here: https://math.stackexchange.com/q/221273/42969 and here: https://math.stackexchange.com/q/169157/42969 and here: https://math.stackexchange.com/q/257907/42969 – Martin R Jun 16 '19 at 16:17

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Note that corners and cusps are out of the question because either would produce a jump discontinuity (of finite length) in the image of $f'$. The remaining options of both a removable discontinuity or a hyperbolic portion are both plausible since they both preserve the notion of at limit at $c$ [keep in mind that the hyperbola would preserve the limit if of odd degree]. This should help you answer your question.

LunarGyrial
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