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.