Prove that a distribution function of a integer valued random variable converges

Question

Let $X$ be integer-valued and let $F$ be it's distribution function. Show that for every $x$:

$P(X=x) = \lim_{\epsilon \downarrow 0}[F(x + \epsilon) - F(x - \epsilon)]$

The intuition is obvious but unfortunately I have no idea how to prove the expression above, so even a hint on how to start would be most welcome!

Answer 1

The statement is true for every random variable $X$.

It is enough to prove that:

• $\lim_{\epsilon\downarrow0}P(X\leq x+\epsilon)=P(X\leq x)$ (continue à droite)
• $\lim_{\epsilon\downarrow0}P(X\leq x-\epsilon)=P(X<x)$ (limite à gauche)

This because $P(X=x)=P(X\leq x)-P(X<x)$

Answer 2

First let $x$ be an integer. If $0<\epsilon <1$ then the event $ \{ x-\epsilon <X \leq x+\epsilon \}$ is same as the event $ \{X=x\}$ because there are no integer points in $(x-\epsilon, x+\epsilon]$ except $x$. Just take probability to see that $F(x+\epsilon)-F(x-\epsilon)=P\{X=x\}$ for all $\epsilon \in (0,1)$. If $x$ is not an integer then we get $F(x+\epsilon)-F(x-\epsilon)=P\{X=x\}=0$ for all $\epsilon $ sufficiently small by a similar argument.