If $\,\lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}\,$ exists for every $x$, what does this imply for $f$?

real-analysis calculus limits derivatives

Let $$g(x,h) = \dfrac{f(x+h)-f(x-h)}{2h},\quad g_0(h) = g(x_0,h).$$ Easy to see that $$g_0(h)=0\ \text{if}\ x_0 = 0,$$ as for any another rational $x_0.$

So the function $g_0(h)=0$ is differentiable in the first case, within $g'_0 = 0.$

In the second case, there is a removable discontinuity of the derivative in the point $x=0$.

In the third case, it can exist the arbitrary (countable) quantity of the removable discontinuities or the gaps of the derivative.

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