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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.

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    In think i got it. Actually, this counterexample is the answer to my question. Thank you very much. I appreciate your help.2012-12-14

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A counterexample to the question you probably mean to ask could be $ G(x;a,c)=\frac1{1+|c\cdot x|}, $ around $b=0$. Then, $G(x;a,0)=1$ but $G(x;a,b)\geqslant3/4$ only if $|x|⩽1/(3|b|)$ hence $G(x;a,λ)\geqslant3/4$ uniformly on $x$ is impossible for each $λ≠0$. A fortiori, $G(x;a,λ)\geqslant3/4$ is impossible uniformly on $x$ and on every $|λ|⩽r(0)$ as soon as $r(0)>0$.