Suppose that $A$ and $B$ are compact subsets of $\mathbb{R}^{n}$. Consider the function $G(x;a,b)$ where $x\in\mathbb{R}^{n}$, $a\in A$ and $b\in B$. We also suppose that $G(x;a,\cdot)$ is uniformly continuous. If it holds that
\begin{align}
G(x;a,b)\geq \delta
\end{align}
is it possible for all $\lambda\in{\cal B}(b,r)\cap B$ (where $r=r(b)>0$), to hold
\begin{align}
G(x;a,\lambda)\geq \frac{3\delta}{4}
\end{align}
where ${\cal B}$ stands for the open ball?
I think the answer is yes and has something to do with the uniform continuity. Any suggestion will be highly appreciated.