Why aren't CDFs left-continuous? Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous:

Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be a sequence of real numbers such that $y_1 > y_2 > \ldots$ and $\lim_i y_i = x$. Let $A_i = (-\infty, y_i]$ and let $A = (- \infty, x]$. Note that $A = \cap_{i=1}^\infty A_i$ and also note that $A_1 \supset A_2 \supset \ldots$. Because the events are monotone, $\lim_i P(A_i) = P(\cap_i A_i)$. Thus,
$$
F(x) = P(A) = P( \cap_i A_i) = \lim_i P(A_i) = \lim_i F(y_i) = F(x^+)
$$

But why doesn't this argument work in reverse to show that $F$ is left-continuous? That is, if we supposed that the $y_i$ were approaching $x$ from the left, why can't we analogously say:
$$
F(x) = P(A) = P( \cup_i A_i) = \lim_i P(A_i) = \lim_i F(y_i) = F(x^-)?
$$

The right continuity of CDFs is a matter of convention. One could just as well take the default to be the left-continuous function $x\mapsto \Bbb P[X<x]$, as one finds in the Russian literature of yesteryear (and perhaps even today); cf. B.V. Gnedenko's Theory of Probability.
