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Is there a standard name for the function: $$ f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases} $$ And is there a potential application of this function? All I can think of is that it will be able to sort ratios according to the "magnitude" of the ratio.

I would see it as a "absolute value" function that deals with the multiplicative identity, since the modulus function can be defined as such: $$ |x| = \begin{cases} 0-x & \text{if x<0;}\\ x & \text{if x≥0;}\\ \end{cases} $$

And $0$ is the additive identity.

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I'm not sure I understand the idea of having such $f$ as an absolute value. An absolute value "with respect to multiplicative identity" sounds like a function that maps $e^x \mapsto e^{|x|}$. – Pedro M. Feb 7 '12 at 12:59
    
I don't know about function names, but a construction similar to that is sometimes useful in the localization of roots of polynomials... – J. M. Feb 7 '12 at 13:06
    
for positive $x$ you can use $$\mathrm e^{|\log{x}|}$$ if you need to include negative value into consideration, then $$\operatorname{sgn}(x)\mathrm e^{|\log|x||} $$ so it is just a use of absolute value function in the composition with other – Ilya Feb 7 '12 at 13:12
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I'm not sure what you mean by "is there a function". – Alex Becker Feb 7 '12 at 13:39
    
"has a standardised name been assigned to the function" – randName Feb 7 '12 at 14:14

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