Thanks in advance for any help. I think I might just be missing something simple.

Problem:

Let f be a continuous function who's domain is the closed interval [0,1] and whose co-domain is also the closed interval [0,1]. Prove that there exists some value c in [0,1] such that f(c)=c .

Hints Provided:

1) consider the function g(x)=x-f(x) . Argue that g is continuous on [0,1] . Draw some inference about g(0) and g(1), specifically, are they positive or negative. Apply Intermediate Value Theorem to the function g.

What I have so far:

F(x) must be equal to x, ie. f(x)=x

0<=c<=1

0<=f(c)<=1

f(0)<=f(c)<=f(1)

Using the above I have proved that both f(c) and c are in the interval [0,1], but not that f(c)=c.

I kind of feel like I am not supposed to assume an actual function for f (not assume that f(x)=x), but I am not sure how to proceed then.

I'm not quite sure where the hint regarding g(x) fits in. If f(x) is in fact x then g(x)=x-x = 0, so neither positive or negative.