I'm taking an intermediate course in probability theory (that is without measure theory) and when defining singular random variables (after showing the devil's function), the book defines:
$X$ is a singular random variable iff $F(x)$ (cumulative distributive function) is continuous and $F'(x)=0$ a.e.
Then it says $X$ is a singular random variable if and only if there exists a lebesgue measure zero set $B$ such that $P(X\in B)=1$ and $F(x)$ is continuous.
It has no proof and I can't find one myself. Could you give me some hints?
Thanks and regards!