Real Analysis: Properties of Continuity

[squaremeplz]

Homework Statement

Suppose f is continuous on [0,2]and thatn f(0) = f(2). Prove that there exists x,y in [0,2] such that |y-x| = 1 and f(x) = f(y)

Homework Equations

The Attempt at a Solution

I got the following 1 line proof.

Suppose g(x) = f(x + 2) - f(x) on I = [0,2]

this proofs that |x - y| = 1 for x = 1, y = 2

and f(x) = f(y)


thanks!

 

[d_leet]

As you have defined g, unless you know more about f, then g is only defined at 0 and no other points in [0,2]. I mean g(1)=f(1+2)-f(1)=f(3)-f(1), but what is f(3)?

And how does what you have written show that f(1)=f(2) certainly there are continuous functions on [0,2] with f(0)=f(2) but that do not satisfy f(1)=f(2).

What do you know about the function h(x)=f(x+1)-f(x)?