# Differentiability of characteristic functions and moments of the corresponding measure

Let $$f$$ be the characteristic function of a real-valued random variable $$X$$. It is known that if $$f$$ has a $$k$$-th order derivative (for some even $$k$$) then $$\mu$$ has a finite $$k$$-th order moment. Is there any reference or discussion the case when $$k$$ is odd? For example, if we only know that $$f$$ has its first order derivative, do we know that $$\mu$$ has finite first moment?