= Non-strict inverse of a continuous nondecreasing function
{title2=$h(a)=\inf\{t:f(t)\geq a\}=G(a-)$}
For a continuous unbounded <nondecreasing function> $f$ on the nonnegative half-line and $a>f(0)$, the non-strict inverse $h$ is left-continuous in its level argument. It equals the left limit of the <strict generalized inverse of a nondecreasing function> $G(a)=\inf\{t:f(t)>a\}$. Indeed $G(b)\leq h(a)$ for $b<a$, while for every $t<h(a)$, $f(t)<a$ and choosing $b\in(f(t),a)$ gives $G(b)\geq t$. Similarly $h(a+)=G(a)$. A flat part of $f$ at level $a$ gives a discrepancy between the two inverses: $h(a)$ selects the beginning of the flat part and $G(a)$ its end. For a <Brownian running maximum>, these are the non-strict and strict passage times.
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