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 $$ d_Y(f(x_1),f(x_2)) \le \varepsilon(d_X(x_1,x_2)). $$