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

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.

• In the first section of my post at stats.stackexchange.com/a/298434/919 (on the distinction between continuous and absolutely continuous variables) I sketch a demonstration.
– whuber
Sep 26, 2021 at 12:48

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 $$\lim_{x\uparrow a}=\Pr\{X.
3. Since $$\Pr\{X\leq a\}=\Pr\{X, 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 $$\cup_{n=1}^\infty \{X\leq x_n\} = \{X, and use the continuity from below of the probability measure $$\Pr(\;\cdot\;)$$.