 Solution

Let $Q=F^{-1}$ be the quantile function. For $p\in (0,1)$,The reverse implication is immediate from the infimum defining $Q$. For the forward implication, if $Q(p)<t$ there is a point $s\le t$ with $F(s)\ge p$. If $Q(p)=t$, choose $s_m<t+1/m$ with $F(s_m)\ge p$; monotonicity and right continuity giveThe endpoint values $p=0,1$ have zero probability under a continuous uniform variable.

ConsequentlyThus the inverse transform sampling identity is