2

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 $t$.

Jakob
  • 21

2 Answers2

4

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 $a=\inf\{x\mid f(x)>t\}$.

Zev Chonoles
  • 129,973
  • 7
    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. – hmakholm left over Monica Dec 01 '11 at 21:16
  • That's a much better suggestion - I've CW'ed the post and added it. – Zev Chonoles Dec 01 '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 01 '11 at 21:30
  • Beautiful, thank you Zev and Henning! – Jakob Dec 01 '11 at 21:56
  • 2
    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 01 '11 at 23:05
  • These are not so much notations as formulas for what's to be denoted. – Mariano Suárez-Álvarez Dec 02 '11 at 08:15
2

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

Mariano Suárez-Álvarez
  • 135,076