# Fixed Point in Continuous Set

1. Nov 20, 2008

### angelpsymon

1. The problem statement, all variables and given/known data
Suppose f:[a,b] $$\rightarrow$$ [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x.

2. Relevant equations

3. The attempt at a solution
I was going to try something with the IVT, but then I realized I wasn't sure what they meant by a fixed point much less how to solve this problem. Any help would be appretiated.

2. Nov 20, 2008

### Unco

Hi Angelpsymon,

It says what a fixed point is in the problem statement: "x such that f(x) = x." You are absolutely correct in thinking to apply the intermediate value theorem. Hint: since f maps into [a,b], we must have that $$f(a)\geq a$$ and $$f(b)\leq b$$.

3. Nov 20, 2008

### Dick

As Unco suggested, maybe without spelling it out completely, apply the IVT to f(x)-x.

4. Nov 20, 2008

### Unco

Apologies, Dick, I certainly didn't mean to do so.

5. Nov 20, 2008

### Dick

You don't HAVE to spell it out completely. Hints are enough. I apologize if I spoiled your hint and made it too obvious. I was just saying how to apply the IVT.

6. Nov 22, 2008

### angelpsymon

Alright, I think that I got it now. Thanks a lot guys.