If $f(x+a)-f(x)$ is differentiable for each $a$, then $f$ is differentiable

Question

$ f \in C(\mathbf{R})$, if for each real $a$, $f(x+a)-f(x)$ is differentiable, then $f$ is differentiable.

It seems hard to convert difference to the original function

Answer 1

This and many other results of this kind are proved in the paper

Bruijn, de, N. G. (1951). Functions whose differences belong to a given class. Nieuw Archief voor Wiskunde, serie 2, 23, 194-218.

The assumption that $f(x)$ is continuous can be weakened, e.g., it's enough to assume that $f(x)$ is bounded on a set of positive measure. Some condition on $f(x)$ is needed since, if $f(x)$ is a discontinuous additive function, then $x\mapsto f(x+a)-f(x)$ is the constant function $x\mapsto f(a)$.