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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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    No measure-theory in there.2012-12-14
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    The answer to the question is YES: choose $\lambda=b$.2012-12-14
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    A counterexample to the question you probably *mean* to ask could be $G(x;a,c)=\exp(-c^2x^2)$ around $b=0$.2012-12-14
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    Listen, you might want to seriously revise your post. I tried to indicate that the existence of **some** $\lambda$ in $B(b,r)$ is trivial since $\lambda=b$ fits the bill. If you want the inequality to hold for **every** $\lambda$ in $B(b,r)\cap B$, please modify the question--and then ponder the example in my other comment.2012-12-14
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    @did, actually i did not understand your example. Could you be more clear please ?2012-12-14
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    In the example I mentioned, $G(x,0)=1$ but $G(x,b)\geqslant\frac34$ only if $|x|\leqslant K/|b|$ for some fixed $K$ hence $G(x,\lambda)\geqslant\frac34$ uniformly on $x$ is impossible for each $\lambda\ne0$, and a fortiori uniformly over every $|\lambda|\leqslant r(0)$ as soon as $r(0)\gt0$.2012-12-14
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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 $r(0)>0$.