Suppose f and g are each defined on an open interval I, a∈I and a∈C(f)∩C(g). If f(a)g(a), then there is an open interval J such that f(x)g(x) for all x∈J.

Question

Suppose f and g are each defined on an open interval I, a∈I and a∈C(f)∩C(g). If f(a)>g(a), then there is an open interval J such that f(x)>g(x) for all x∈J.

anonymous · Answer

what is C(f) ?

studybird · Answer

The set of all points at which f is continuous is denoted C(f)

anonymous · Answer

@studybird: Do you usually use the epsilon/delta definition or the topological definition of continuity? Both definitions could be used to prove the above statement...

studybird · Answer

we've been covering delta epsilon but lately we've been doing topology

anonymous · Answer

I'll give a rough informal outline of a proof as I'd probably try it in the epsilon/delta style...

Both, f and g are continuous at a. This means that you can choose an open interval around a in which f will not change more than a given small amount (let's call this amount $\epsilon$). The same is true for g. If your choose you $\epsilon$ small enough (e.g. one third of the difference between f and g at a) you can be sure that within the smaller of the two open intervals the graphs of the two curves will not "touch".

Within such an interval (let's call it $a-\delta/2, a+\delta/2$) you can be sure then that the graph of f will stay "above" g which is what you want to prove, right?

I propose you now try to transform this prove into a more formal argument if you agree that it solves your problem...

anonymous · Answer

Typo: There's the left bracket missing from the interval around a. Guess you figured that already. ;-)

anonymous · Answer

Jerico can u help me

anonymous · Answer

@jatinbansalhot: Please send me a message with a link to your question so that this question remains on topic.

studybird · Answer

OK got it, thanks!