Functions that have no first order derivative might have fractional derivatives of all orders less than one (Q1890609)

The main question studied in the paper is whether there exists a function \(f(x)\), \(x\in \mathbb{R}^ 1\), which everywhere has a continuous fractional derivative \(D^ \alpha\) of any order \(\alpha< \alpha_ 0\) but nowhere has the fractional derivative just of order \(\alpha_ 0\). The answer is positive and it is shown that the well-known Weierstrass function \(\sum^ \infty_{n= 0} q^{-\alpha_ 0 n} e^{iq^ n x}\), \(q> 1\), \(\alpha_ 0> 0\), may be used as such an example. The Riemann function \(\sum^ \infty_{n= 1} {\cos n^ 2 x\over n^ 2}\) is also studied from this point of view.