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What operations is a metric closed under?
Say we have a metric $d: X \rightarrow \mathbb{R}$, and a function $f(d)$ that takes in a metric $d$ and (ideally) spits out another metric.
As an example, let $d$ be a metric and consider the function $f$ where $f(d) = \sqrt{d}$ - then the output $\sqrt{d}$ is also a metric. Similarly, if $f(d) = \frac{d}{1+d}$ then again the output $\frac{d}{1+d}$ is also a metric.
I am wondering if we can say something about what properties of $f$ are necessary / sufficient in order to ensure that the output will be a metric.
Please feel free to make any changes / suggestions the problem since I'm very very new to analysis / functional analysis.