Cumulative distribusion function continuous if and only if P(X=x)=0?

Question

In my lecture notes for my probability course it is said that for random variable $X$ with cdf $F(x) = P(X \leq x)$, we have $$F(x) \text{ is continous at all } x \Longleftrightarrow \forall x \in \mathbb{R},\; P(X = x) = 0.$$

This makes intuitive sense, but would trivialise some one of my set questions, can someone prove/disprove it or give a textbook reference?

Answer 1

$F$ is right-continuous. If $F(x-)=\lim_{ y<x, y \to x} F(y)$ then $F$ is continuous at $x$ iff $F(x)=F(x-)$. But $F(x-)=P(X <x)$ so $F(x)-F(x-)=P( X=x)$. Hence $F$ is continuous at $x$ iff $F(x)-F(x-)=0$ iff $P(X=x)=0$.