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Let $F: \mathbb{R} \to [0, 1]$ be a distribution function (so it is right continuous, non-decreasing and $\lim_{x \to \infty} F(x) = 1$, $\lim_{x \to -\infty}F(x) = 0$.

When is $\inf \{x \ | \ F(x) > a\} \neq \inf \{x \ | \ F(x) \geq a\}$ for some $a \in (0, 1)$?

I have trouble of even thinking of one example. If we take $F$ to be continuous, then $\{x \ | \ F(x) > a \} = F^{-1}([a, \infty)) = (F^{-1}(a), \infty)$ and $\{x \ | \ F(x) \geq a \} = [F^{-1}(a), \infty)$, so in that case the infima are always equal.

Say we take $F$ to have a discontinuity at $a$, say $F(x_0) = a$, $\lim_{x \uparrow x_0} F(x) = b < a$, the infima at $a$ are $x_0$ in both cases.

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