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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.

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    $\begingroup$ 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. $\endgroup$
    – whuber
    Sep 26, 2021 at 12:48

1 Answer 1

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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 $\Pr(\;\cdot\;)$.

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