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?
A good reference regarding this type of results is the book by Eugène Lukacs "Characteristic Function". For example, chapter 2.3 "Characteristic functions and moments" provides results in this direction. Theorems 2.3.1, 2.3.2 and 2.3.3 might be what you are looking for.