$X$ is a random variable taking values in $\mathbb{R}^d,||.||$ an arbitrary norm on $\mathbb{R}^d.$
Prove that the characteristic function $\phi_X$ is differentiable at $0$ if and only if $\lim_{x \to +\infty}xP(||X|| \geq x)=0$ and $\lim_{x \to +\infty}E[X1_{\{||X||<x\}}]$ exists in $\mathbb{R}^d$.
Is it possible to generalize this condition to obtain the differentiability at any point $x_0 \in \mathbb{R}^d?$
Any ideas how to do it? Because it doesn't seem it's an obvious problem.