Need help on this proof

• Oct 22nd 2009, 11:31 PM
binkypoo
Need help on this proof
I have to prove the following:

Suppose that functions f and g are continuous at $x=c\in (a,b)$ and f(c) > g(c). Prove there exists $\delta$ > 0 such that for all $x\in (a,b)$ with $|x-c|<\delta$, we have f(x) > g(x).

I really have no idea. any help on this would be great thanks!
• Oct 23rd 2009, 12:53 AM
el_manco
Hello

First you can proof this result.

If $f:(a,b)\longrightarrow R$ is a continuous function in $c\in (a,b)$ then and $f(c)>0$, then there exists $\delta>0$ such that $|x-c|<\delta, x\in (a,b)$ implies $f(x)>0$.

This follows inmediately from continuous definition; taking $\epsilon=f(x)$ there exists $\delta>0$ such that $|x-c|<\delta, x\in (a,b)$ implies:

$|f(x)-f(c)|-f(c)\quad \Rightarrow\quad f(x)>0$

Then, for your exercise, apply this result to the funcion $f-g$. Note that the difference of continuous function is continuous.

Best regards.
• Oct 25th 2009, 09:18 PM
binkypoo
[quote=el_manco;389581]taking $\epsilon=f(x)$
do you mean $\epsilon=f(c)$??
• Oct 26th 2009, 12:40 AM
el_manco
[quote=binkypoo;391521]
Quote:

Originally Posted by el_manco
taking $\epsilon=f(x)$
do you mean $\epsilon=f(c)$??

Yes, of course. I meant $\epsilon=f(c)$. Sorry for the typo.

Best regards.