Is the characteristic function of a random variable continuously differentiable?

Question

I know that the characteristic function $\varphi(t) = E[e^{itX}]$ of a random variable $X$ is uniformly continuous. But is it continuously differentiable? I can bound its derivative $$ \lim_{h \to 0}\left|\frac{\varphi(t+h) - \varphi(t)}{h}\right| \le E[|X|] $$ Hence I think it might not be defined when $E[X] = \infty$. But I am having a hard time coming up with a counter-example. Can someone give me a hint? Thank you very much!

Answer 1

The characteristic function of the Cauchy distribution is $e^{-|t|}$ which is not differentiable at $0$.