[Math] Quantile function continuity

Question

Given an increasing, right continuous function $$F:\mathbb{R} \rightarrow [0,1]$$ such that $\lim_{x\rightarrow \infty} F(x)=1 $ and $\lim_{x\rightarrow -\infty} F(x)=0$

then we can define $G:(0,1)\rightarrow \mathbb{R}$ $$G(y)= \inf\{x : F(x)\geq y\}$$

I could show that $G$ is increasing and that $F(x) \geq y$ is equivalent to $x \geq G(y)$.

How can I prove that G is also left-continuous?

Any help would be appreciated!

Answer 1

Hint: Monotone functions may have only jump discontinuities. Prove that $\sup_{y<y_0}G(y)=G(y_0)$.

Let $x' := \sup_{y<y_0}G(y)$. Then for each $y<y_0$ and $\epsilon>0$ we have $F(x'+\epsilon)>y$. Right continuity of $F$ implies $F(x')\ge y$. Hence $F(x')\ge y_0$.