Strongly \(\mathbb Q\)-differentiable functions (Q1852439)

The authos calls a real function \(f:\mathbb R\to\mathbb R\) strongly \(\mathbb Q\)-differentiable if the finite limit \[ D_h^{\mathbb Q}f(x_0):=\lim_{\mathbb R\times\mathbb Q^+\ni(x,r)\to(x_0,0)} {{f(x+rh)-f(x)}\over{r}} \] exists for all points \(x_0\in\mathbb R\) and direction \(h\in\mathbb R\). It is obvious that functions that are either continuously differentiable or additive are always \(\mathbb Q\)-differentiable. The main result of the paper offers the more surprising statement that a \(\mathbb Q\)-differentiable function can always be decomposed as the sum of a continuously differentiable and an additive function.