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Consider the function

$$f(x) = e^{-\lvert\log x\rvert} = \begin{cases} x & \text{ if } |x| \leq 1,\\ x^{-1} & \text{ otherwise} \end{cases}$$

This might appear in an argument of the following nature, for instance:

Suppose that $\psi$ is a continuous function such that $\psi(c x) = \psi(x)$. Then $\psi(x) = \psi(f(c) x) = \psi(f(c)^2 x) = \dots = \psi(f(c)^n x)$ for all $x$. Since $\psi$ is continuous, we have $$\psi(0) = \psi(\lim_{n \to \infty} f(c)^n x) = \lim_{n \to \infty} \psi(f(c)^n x) = \psi(x)$$ for all $x$. So $\psi$ is constant.

Does this function have an existing name? (Bonus question: is there a name for the function $g(x) = -|x|$?)

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    I can't say that there isn't a paper where someone have called these functions with a special name, however I can say that there is no established (widely known) name for this that you can assume people will know. And that is what really matters: if you say "where $f(x)$ is the xyzw function" without defining the xyzw function just assuming that it's known then this just causes confusion rather than clarity. – 2017-01-01
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    @Winther: I find it's often the case that there are names for things that I'd understand if I saw them, but which don't occur to me off the top of my head. For instance, you might write something like $\binom{n}{k}$ all the time without consciously referring to it as a "binomial coefficient," but when people say "binomial coefficient" you know precisely what they mean. – 2017-01-01
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    Turns out a very close variant of this has been asked before, see [What is the name of this function?](http://math.stackexchange.com/questions/41171/what-is-the-name-of-this-function) – 2017-01-01
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    $\min(x,1/x)$, maybe? – 2017-01-01

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