Notation for forcing a real value between bounds

Question

I am simply interested in ways to write $$ x \mapsto \min(b, \max(a, x)) $$ in a more intuitive way for some $a<b \in \mathbb{R}$.

Something like $x \mid_a^b$ but not something I invented just now.

Best one so far: $ a \lor x \land b $ thanks to HallaSurvivor.

Answer 1

Some people write $a \wedge b$ for the min function, and $a \vee b$ for the max. Then the function you want would be written

$$x \mapsto a \vee x \wedge b$$

Or, even more succinctly

$$a \vee - \wedge b$$

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I hope this helps! ^_^

Answer 2

How about the following? :) $$x \mapsto \max(a, \min(b, x))$$

But seriously, $$x \mapsto \begin{cases} a &\text{if $x<a$}\\ x &\text{if $a \le x \le b$}\\ b &\text{if $x>b$} \end{cases} $$

Answer 3

Perhaps the most direct and clear way is simply to split into cases: $$ x\mapsto\cases{a& if $x<a$\\x& if $a\leq x\leq b$\\b& if $b<x$} $$ But it can also be done with absolute values: $$ x\mapsto \frac{|x-a| -|x-b|}{2} +\frac{a+b}2 $$ However, this is probably not the most transparent way of writing it.