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Given real numbers $a$ and $b$ satisfying $a \leq b$, define:

$$\langle a,b\rangle (x) = \mathrm{min}(b,\mathrm{max}(a,x)) = \mathrm{max}(a,\mathrm{min}(b,x))$$

(These numbers are equal because $a \leq b$. See here for some relevant abstract generalities.)

So basically, the function $$\langle a,b\rangle : \mathbb{R} \rightarrow \mathbb{R}$$ has the effect of "truncating" a real number so as to lie in the interval $[a,b]$.

This function shows up implicitly in one of my applied math homework assignments. We're modelling the amount of fluid in a reservoir; due to rainfall or evaporation, the volume changes, but it always remains between $0$ and $V_\mathrm{max}$, and hence the function $\langle 0,V_\mathrm{max}\rangle$ implicitly shows up in our numerical model.

I'll bet the function $\langle a,b\rangle$ shows up all over applied math, so:

Question. Does this function have an accepted name or notation?

goblin GONE
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  • In programming, we usually call that a "constrain" function. I have never seen such a function in mathematical texts. Usually, a mathematician will simply state that $x$ lies on the interval $[a,b]$. Mathematics is less structured than programming, so we don't need functions for everything. – pseudoeuclidean Apr 15 '16 at 03:09
  • @pseudoeuclidean, I don't much like that style of mathematics in which the author(s) can't be bothered to make explicit the functions they're implicitly using. It's low quality, and deserves our criticism. – goblin GONE Apr 15 '16 at 03:23
  • My mathematics teachers have always told me, "mathematicians are the laziest people alive". My teachers would agree with you. However, implicitly defining a problem is often less restrictive in a way. It frees the mind to think about the problem in any way that you'd like, without a strict, explicitly defined structure. – pseudoeuclidean Apr 15 '16 at 03:28
  • @pseudoeuclidean, I can agree with that. I'm not averse to mixing in a bit of vague "premathematics" into the actual math e.g. to help the reader get onto the same page as you. But we shouldn't leave it at that, despite that this is so often done. – goblin GONE Apr 15 '16 at 03:31

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