Prove that a distribution is continuous at a point if and only if it has zero probability at that point

Question

Let $X$ be a random variable with distribution function $F$. Prove that $F$ is continuous at $x = a$ if and only if $\mathbb{P}(X = a) = 0$.

Could anyone give some hints? I am wondering where I should start with.

Answer 1

Try this:

1. Read this answer carefully to understand how to prove the right continuity of $F$.
2. With the same style of reasoning used in that proof, try proving that $\lim_{x\uparrow a}=\Pr\{X<a\}$.
3. Since $\Pr\{X\leq a\}=\Pr\{X<a\} + \Pr\{X=a\}$, putting it all together, it follows that $F$ is continuous at $a$ if and only if $\Pr\{X=a\}=0$.

Hint for the second item: pick a strictly increasing sequence $\{x_n\}_{n\geq 1}$ with limit $a$, prove that $\cup_{n=1}^\infty \{X\leq x_n\} = \{X<a\}$, and use the continuity from below of the probability measure $\Pr(\;\cdot\;)$.