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What is the name of the following function (if there is one)?

$$f(x) = \begin{cases} x & \text{ if } -1 \leq x \leq 1\\ \frac 1x & \text{ if } x < -1 \text{ or } x >1\\ \end{cases} $$

If this function has a name, how is it usually denoted?

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    Perhaps the context where you came across this might help...2011-05-25
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    I've never seen the function in any particular context. I was trying to find a function so that for numbers $a$ and $b$ greater than 0: $f(\frac{a}{b})$ is always less than or equal to 1; $f(\frac{a}{b})$ is largest when $a=b$; $f(\frac{a}{b})$ is increasing when b>a; $f(\frac{a}{b})$ is decreasing when a>b.2011-05-25
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    Why would you expect it to have a name? It's equal to $\min(1, x^2)/x$ (except for $x=0$).2011-05-25
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    This function seems like it would be useful in many cases. Another way of writing the function: For numbers $a$ and $b$, $f(\frac{a}{b})=\text{min}(\frac{a}{b},\frac{b}{a})$. Isn't it often that a ratio less than one is needed, rather than the ratio's inverse?2011-05-25
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    $\mathrm{sgn}(x)e^{-|\log(|x|)|}$ ?2012-07-08

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