Let $(A,\rho)$ be a compact metric space and let $f: A \to A$ be a function satisfying $ \forall \quad x,y\in A, \ x \ne y \ \ \implies \rho(f(x),f(y))< \rho(x,y). $
Now define the function $\chi: A \to \mathbb{R}$ by $\chi(x) = \rho(x,f(x)), \qquad x \in A.$
Then I have to show that $\chi$ is uniformly continuous.
I have to show for every $\epsilon >0$, there exists $\delta>0$ such that for every $x,y \in A$ with $\rho(x,y) < \delta$ we have $\rho(\chi(x),\chi(y)) <\epsilon$.
Can anyone explain in detail?