Counter-example: f(x) = x + 1
The theorem is true if f is bounded and continuous on R. To prove it, consider the function g(x) = f(x) - x. Use the fact that f is bounded and try to show that g has at least one root.
A recent calc problem assigned to me was:
Suppose that f is a continuous function on [0,1]. If 0 < f(x) < 1 for all x E[0,1] show that there is a point c such that f(c) = c.
I figured I'd be using the intermediate value theorem at least, but I can't figure out how to prove that a point c would produce a function value of c. It seems to me that that wouldn't be the case. Anyways, any help would be much appreciated. Thanks!