# Notation for “first $x$ where $f(x)$ greater than $t$”?

What is the preferred notation for expressing the first $x$ where $f(x)$ is greater than a threshold $t$. This is similar to $\arg\max$ notation but instead of max, I want the first $x$ where $f(x)$ is greater than $t$.

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I don't think there's a standard notation for this concept specifically, but one could express it as $$\inf f^{-1}((t,\infty)).$$ Alternately, as Henning points out below, we could write $$\min\{x\mid f(x)>t\},$$ which is indeed significantly clearer.

Note that using $\inf$ instead of $\min$ guarantees the quantity exists, but we may not actually have $f(a)>t$ where $a=\inf\{x\mid f(x)>t\}$.

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The open interval notation inside parentheses are confusing, I think. It would be worth the extra clarity to write it out explicitly as $\min\{x\mid f(x)>t\}$ -- which is not even that much longer. – Henning Makholm Dec 1 '11 at 21:16
That's a much better suggestion - I've CW'ed the post and added it. – Zev Chonoles Dec 1 '11 at 21:24
If interval notation is to be used, a better (= more readable) version is $\inf f^{-1}[(t,\to)]$ (or $\inf f^{-1}[(t,\infty)]$, though I prefer the arrow form). If $R$ is a binary relation and $S$ is a subset of its domain, I much prefer $R[S]$ to $R(S)$ for $\{y:\exists x\in S(x R y)\}$. – Brian M. Scott Dec 1 '11 at 21:30
Beautiful, thank you Zev and Henning! – Jakob Dec 1 '11 at 21:56
This comes up all the time in probability (usually $f$ is a stochastic process) and is almost universally denoted by $\inf\{x : f(x) > t\}$ (or $\min$ if appropriate). So +1 for @HenningMakholm. – Nate Eldredge Dec 1 '11 at 23:05

My prefered notation for the first $x$ where $f(x)$ greater than $t$ is “the first x where f(x) greater than t”.

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