Consider a continuous function $\phi: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}_{\geq 0}$ and a locally bounded function $\psi:\mathbb{R}^n \rightarrow \mathbb{R}^m$.
So we study functions of the kind $\phi(x,\psi(y))$ for $x,y \in \mathbb{R}^n$.
Prove the following proposition.
Fixed $\bar x \in \mathbb{R}^n$, for any $\epsilon>0$ there exists $\delta > 0$ such that
$ \max_{(x,y) \in X \times X} \{ \phi(x,\psi(y)) - \phi(y,\psi(y)) \} < \epsilon $
where $X = {\bar x} + \delta \mathbb{B}$ is the closed ball around $\bar x$.