Where to find a proof of theorem which says that: if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesque measure or on the set of second category with Baire propert and $$ \frac{\Delta_h^nf(x)}{h^n} \rightrightarrows g(x) \textrm{ as } h \rightarrow 0 $$ on every compact interval $[c,d]$, to a bounded function $g: \mathbb{R} \rightarrow \mathbb R$, then $f$ is of class $C^n$ and $g^{(n)}=f$.

MR1245559 (94k:26024) Frölicher, Alfred(CHGENVSM); Kriegl, Andreas(AWIEN) Differentiable extensions of functions. (English summary) Differential Geom. Appl. 3 (1993), no. 1, 71–90. The review: A condition is given for functions defined on an arbitrary subset of the real line with values in a Fréchet space (or even in a space of more general type) to admit a smooth extension. This condition is that the difference quotients of the corresponding orders are to be locally bounded. The results are obtained for $C^\infty$extensions as well as for finite order extensions with locally Lipschitz derivatives. In the last case a continuous linear extension operator is constructed. 

