Inversion Lemma (Quantiles)

Let $F$ be a CDF and $p\in (0,1)$ and $t\in \mathbb{R}$.

$F(t)\ge p⟺t\ge Q_F^{-}(p),F(t-)\le p⟺t\le Q_F^{+}(p)$

If $F$ is piecewise constant additionally:

$F(t)>p⟺t\ge Q_F^{+}(p),F(t-)<p⟺t\le Q_F^{-}(p)$

$$\begin{array}{rl}F(t)& =\sup \{p\in (0,1)\mid F(t)\ge p\}=\sup \left\{p\in (0,1)\mid t\ge Q_F^{-}(p)\right\}\\ F(t-)& =\inf \{p\in (0,1)\mid F(t-)\le p\}=\inf \left\{p\in (0,1)\mid t\le Q_F^{+}(p)\right\}\end{array}$$