I want to show that $X$ with $\mathbb P(X=1)=\mathbb P(X=-1)=\frac12$ is not infinitely divisible.
This means I need to show that there $\exists n\in\mathbb N$ such that $\nexists X_1,\dots,X_n$ i.i.d such that $\sum\limits_{i=1}^n X_i \overset d= X$. I want to show that for $n=2$. The discrete case is simple. If $X_1, X_2$ assumes one value $X_1+X_2$ assumes also one value. So $X_1, X_2$ must assume at least two values but then $X_1+X_2$ assumes three values. The rest follows from induction.
How about the continuous case? I believe $\varphi_X(t)=\cos(t)$. What argument can I use to prove that the $X_1, X_2$ can't be continuous?
If $U$ and $V$ are independent with at least one of them continuous then $U+V$ is also continuous. Hence, $X$ cannot be the sum of independent r.v.'s with one (or both) of them continuous.
Alternative proof: The characteristic function of an infinitely divisible distribution has no zeros.