= Strict generalized inverse of a nondecreasing function
{title2=$G(a)=\inf\{t\geq0:f(t)>a\}$}
For an unbounded nondecreasing function $f:[0,\infty)\to\mathbb R$, its strict generalized inverse is $G(a)=\inf\{t\geq0:f(t)>a\}$. It is nondecreasing and right-continuous in $a$. If $a_n\downarrow a$, then for any $t>G(a)$ monotonicity gives $f(t)>a$, so eventually $f(t)>a_n$ and $G(a_n)\leq t$; also $G(a_n)\geq G(a)$. Local finiteness gives finite left limits. Flat parts of $f$ create jumps of $G$. For the continuous running maximum of <Brownian motion>, this gives the <càdlàg> <Brownian first-passage subordinator>.
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