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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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    @Ilya your function should be posted as an answer, looking at https://en.wikipedia.org/wiki/Absolute_value#Definition_and_properties it's interesting to see some dualities2016-10-30

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