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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  α  ∈ (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 is not sufficient if the Caputo derivatives have a constant sign for a few values of  α  ∈ (0, 1) only.

Monotonicity of functions and sign changes of their Caputo derivatives