Show that the mean of a stable distribution exists

Question

According wiki, the ch. f. of a $\alpha-$stable distributions is:

$$\varphi(t; \alpha, \beta, c, \mu) = \exp \left ( i t \mu - |c t|^\alpha \left ( 1 - i \beta sgn(t) \Phi \right ) \right ) $$

where $sgn(t)$ is just the sign function and $$ \Phi = \begin{cases} \tan \left (\frac{\pi \alpha}{2} \right) & \alpha \neq 1 \\ - \frac{2}{\pi}\log|t| & \alpha = 1 \end{cases} $$

I would like to show

1. If $\alpha>1$, the mean exists and is equal to $\mu$;
2. Why the mean does not exist if $0<\alpha \leq 1$?

I could deduce the first moment by differentiating the characteristic function, but the sign function is not differentiable at zero.

Helps!