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

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

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