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Let $h>0$. Consider the finite difference formula $$ \frac{f(x+h)-f(x-h)}{2h} \approx f'(x) $$ which is called central difference. It is well known that for a sufficiently smooth function $f(x)$,

$$ \Bigg|\frac{f(x+h)-f(x-h)}{2h}-f'(x)\Bigg| = \mathcal{O}(h^2) $$

I have the following question: If

$$ \lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} $$ exists. Then $$ f'(x) $$ exists?. In other words, is the following implication true? $$ \lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h} \ \text{exists} \implies f'(x) \ \text{exists} $$

I think that it's not necessarily true, but I haven't found a counterexample. Thanks in advance.

Brian
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    Hello and welcome to math.stackexchange. This is a nice question. To get some insight, consider the fact that nothing here is assumed about $f(x)$. So this could be anything. – Hans Engler Jan 31 '23 at 15:30
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    Try $f(x)=|x|$ at $x=0$. – Gerd Jan 31 '23 at 15:42
  • The characteristic function of $\Bbb{Q}\subset\Bbb{R}$ has a symmetric derivative at every rational, but $f$ is discontinuous everywhere… – jp boucheron Jan 31 '23 at 17:54

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