Thread: show that f has a fixed point

If f:[a,b]-->[a,b] is continuous, shwo that f has a fixed point; that is, show that there is some point x in [a,b] with f(x)=x.

I'm really struglling with the continuous functions section with metric space, any help will be appreaciated!

Fixed Point Theorem -- from Wolfram MathWorld

If f(a)= a we are done. If not, then, because f(a) must be in [a, b], f(a)> a.
If f(b)= b we are done. If not, then, because f(b) must be in [a, b], f(b)< b

Let .