Proving Existence of Fixed Points in Continuous Sets

[angelpsymon]

Homework Statement

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.

Homework Equations

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.

 

[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 $$f(b)\leq b$$.

 

[Dick]

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

 

[Unco]

As Unco suggested, maybe without spelling it out completely

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

 

[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.

 

[angelpsymon]

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