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 $F^{-1}(y)=x$?