# How can I show this property about the quantile function?

Let $$F:\mathbb{R}\rightarrow [0,1]$$ be a non-decreasing right continuous function, and it's generalized inverse function $$F^{-1}:[0,1]\rightarrow \mathbb{R}$$ defined as $$F^{-1}(y)=\inf \{x\in \mathbb{R}:F(x)\geq y\}$$.

How can I show that if $$F^{-1}$$ is continuous at $$y$$ and if $$F(x-)\leq y\leq F(x)$$ then $$F^{-1}(y)=x$$?

• If you take $F$ left continuous, why $F(x-)$ and $F(x)$ differ? – NCh Dec 6 '18 at 1:19
• I edited the post, I meant right continuous, my bad, thank you. – Enzo Giannotta Dec 6 '18 at 16:22