I have to prove the following:
Suppose that functions f and g are continuous at and f(c) > g(c). Prove there exists > 0 such that for all with , we have f(x) > g(x).
I really have no idea. any help on this would be great thanks!
First you can proof this result.
If a,b)\longrightarrow R" alt="fa,b)\longrightarrow R" /> is a continuous function in then and , then there exists such that implies .
This follows inmediately from continuous definition; taking there exists such that implies:
Then, for your exercise, apply this result to the funcion . Note that the difference of continuous function is continuous.