Let $f,g: M \rightarrow \mathbb{R}$ be continuous in $a \in M$. If $f(a)<g(a)$, $ \ \exists \delta>0$ such that, for all $x,y \in M$ $$d(x,a)<\delta, d(y,a)<\delta \rightarrow f(x)<g(y) $$.
I Just can't find the right $\delta$ for the problem.
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1Is that all of the information that you have for the problem? – JacobCheverie May 08 '20 at 11:12
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What is the question? – SescoMath May 08 '20 at 11:14
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I assume it is that "If $d(x, a) < \delta$ and $d(y, a) < \delta$, show that $f(x) < g(y)$". All the conditional statement is saying is that both $x$ and $y$ are contained in a $\delta$ neighborhood about $a$. I would think we would need to know more about $f,g$? – JacobCheverie May 08 '20 at 11:16
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That's all the question says. – Lucas Cantü May 08 '20 at 11:22
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Let $\epsilon =\frac {g(a)-f(a)} 2$. There exist $\delta >0$ such that $|f(x)-f(a))| <\epsilon$ if $d(x,a) <\delta$ and $|g(y)-g(a)| <\epsilon$ if $d(y,a) <\delta$. Now if $d(x,a) <\delta$ and $d(y,a) <\delta$ then $f(x) <f(a)+\epsilon =g(a)-\epsilon <g(y)$.
Kavi Rama Murthy
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