Proving the Existence of f(x)=x: A Continuos Function Problem | [0,1]

[jmich79]

Homework Statement

Suppose that f is ais continuos function defined on [0,1] with f(0)=1 and f(1)=0. show that there is a value of x that in [0,1] such that f(x)=x. Thank You.

 

[NateTG]

This is slightly simpler:
Let's say I have a function g:[0,1] \rightarrow \Re, which is continuous, with g(0)=1 and g(1)=-1 can you show that there is an x \in [0,1] so that g(x)=0?

 

[jmich79]

Im Still Not Following. Can You Explain It A Little Bit Better To Me. Thank You For Your Post By The Way.

 

[Office_Shredder]

If a function that's continuous is negative at one point, and positive at another point, does it necesarily cross the x-axis (i.e. is zero somewhere in between)?

That's what he's driving at, but puts it in terms that are more obviously applicable to the problem at hand

 

[Dick]

Define the function G(x)=f(x)-x. Now four questions. i) is G continuous? ii) What are G(0) and G(1)? iii) What does it mean if G(x)=0 in terms of f? iv) Might this have something to do with the NateTG's and Office_Shredder's hints?