Monotonicity of functions and sign changes of their Caputo derivatives

Abstract

It is well known that a continuously differentiable function is monotone in an interval $[a,b]$ if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders $\alpha \in (0,1)$ with starting point $a$ of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it not sufficient if the Caputo derivatives have a constant sign for a few values of $\alpha \in (0,1)$ only.