= Left continuity of the quantile function
{title2=$Q(u)=\sup_{v<u}Q(v)$}
For $Q(u)=\inf\{x:F(x)\geq u\}$, $0<u<1$, the <quantile function> is a <monotone function> and is left-continuous. The <distribution function> is right-continuous, giving $Q(u)\leq x$ exactly when $u\leq F(x)$. 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.
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