# When are two infima of almost the same set not equal.

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.