Left continuity of the quantile function ($Q(u)={\sup }_{v<u}Q(v)$)

For $Q(u)=\inf \{x:F(x)\ge u\}$, $0<u<1$, the quantile function is a monotone function and is left-continuous. The distribution function is right-continuous, giving $Q(u)\le x$ exactly when $u\le F(x)$. If ${\sup }_{v<u}Q(v)<Q(u)$, a point between them has $F(x)<u$ and contradicts this equivalence for some $v\in (F(x),u)$. A jump just to the right of $u$ is possible when $F$ is flat on an interval.