Proving the Existence of a Fixed Point using the Intermediate Value Theorem

[icystrike]

Homework Statement

Let f : [0; 1] -> [0; 1] be continuous on [0; 1]. Prove that there exists C $$\epsilon [0; 1]$$ such
that f(c) = c.

Homework Equations

The Attempt at a Solution

I've manage to prove this by having an extra cont. function g(x)=f(x)-x .. But I am looking for easier proof

 

[micromass]

Sorry, that's the easiest proof available

 

[icystrike]

Sorry, that's the easiest proof available

Is it one of the common proof and example when they conduct lesson on IVT?

 

[micromass]

Yes, this is one of the common examples when discussing the IVT. There are other proofs however, but these are quite complicated (see Brouwers fixed point theorem)

 

[icystrike]

Thank you! Greatly appreciate ur time =D