Thread: Help with Continuity Proof

I think I kinda know the direction this proof needs to go but I'm not sure how to word it all. Any help would be appreciated.

Let be continuous. Prove that there is a , such that [Hint: consider

I'm thinking a proof by contradiction might be apprpriate here. Assume that there is no c that works and then find a contradiction by using the hint somehow????

For which x in [0,1] g(x)>0, for which x in [0,1] g(x)<0 ?

Now use Intermediate value theorem - Wikipedia, the free encyclopedia

Suppose that for all and consider . Since the denominator is non-zero this is continuous and surjective. But, this implies that is connected....