Identity at Some Point

1. Dec 12, 2005

"Let f[a,b]->[a,b] be continuous. Prove that there exists at least 1 x in [a,b] such that f(x)=x."

This seems simple geometrically since if we consider the identity function g(x)=x, if f(x) is continuous, then if you "draw" the graw of f, it must intersect g at some point. At that point, f(x)=x. But I have no idea how to translate this intuition into analytical lingo.

2. Dec 12, 2005

Muzza

Use the intermediate value theorem.

3. Dec 12, 2005

math-chick_41

and what can you say about g(x) = f(x)-x on [a,b]?

4. Dec 12, 2005

HallsofIvy

Staff Emeritus
In particular, what is f(a)- a? What is f(b)-b?
(Remember that f(a) and f(b) is in [a,b]?)

5. Dec 13, 2005