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I think I proved the following theorem, but the proof is pretty long. So I thought maybe there is a reference for it because the statement seems standard. Does anyone know such a reference? Thank you!

Theorem: Let $(X,d_X), (Y,d_Y)$ be metric spaces, $X$ be convex, and $f \colon X\to Y$ be uniformly continuous. Then there is a function $\varepsilon\colon [0;\infty) \to [0;\infty)$, which is concave, continuous in $0$ and has the properties that $\varepsilon(0) = 0$ and, for all $x_1,x_2 \in X$, it is $$ d_Y(f(x_1),f(x_2)) \le \varepsilon(d_X(x_1,x_2)). $$

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    $\begingroup$ Sorry, I found it finally: en.wikipedia.org/wiki/… I just didn't know the appropriate words to search for it. $\endgroup$ – Kolodez Apr 30 '18 at 17:32

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