Proof that there is a solution to f(x)=x

[monkeybird]

This is my first analysis class and I'm really confused. My attempt may not make much sense because I just followed an example in my book, so any help would be greatly appreciated. Thanks!

Homework Statement

Suppose that f: R -> R is continuous and that its image f(R) is bounded. Prove that there is a solution of the equation f(x) = x, x in R.

Homework Equations

Intermediate Value Theorem.

The Attempt at a Solution

Assume there is a solution to the function f. Observe that f(-1)<0 and f(1)>0. Since f is a continuous function, by applying the Intermediate Value Theorem to the restriction f:[-1,1] -> R, we can conclude that the point x0 in the open interval (-1,1) is a solution of the function f.

 

[snipez90]

I think you've misunderstood the question. You're trying to find a fixed point, which is basically a point such that f(x) = x, where x is in the domain of some function f. This means that your function is not the identity function f where f(x) = x, but rather all you are told is that f is continuous and f(R) is bounded and for all such functions you need to show that f(x) = x for at least one x. Does that make sense?

Also, you can't start your proof with assuming a solution exists. You need to actually use a theorem, such as the intermediate value theorem, to demonstrate existence. My advice would be to confine yourself to the domain [0,1], draw some continuous function, and see why your function must have a point where f(x) = x.

 

[Gavins]

Define a new function as f(x) - x and then try using the IVT.