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What operations is a metric closed under?
Let $f: \mathbb R_{\geq 0} \to \mathbb R_{\geq 0}$ be a function and $d : X \times X \to \mathbb R_{\geq 0}$ a metric.
I've been thinking about what properties $f$ has to have for the following to hold: $f(d(x,y))$ is a metric if and only if $f$..., that is, I am looking for a necessary and sufficient condition on $f$ such that $f \circ d$ is a metric.
Clearly, we have to have $f(0) = 0$ since otherwise $d^\prime (x,y) := f(d(x,y))$ doesn't satisfy $d^\prime (x,y) = 0$ if and only if $x=y$. Also, we want that the only point that is mapped to zero by $f$ to be zero.
Since we always have $d^\prime (x,y) = d^\prime (y,x)$, all that remains to think about is the triangle inequality. At first I thought that $f$ had to be decreasing, i.e. $x