# F(x)=x for some x

## Homework Statement

I have to prove that if a function is bounded for x between (0,1) and f(x) is bounded also between (0,1) and f(x) is continuos that there exists some x such that f(x)=x

## Homework Equations

Intermediate value theorem?

## The Attempt at a Solution

I know that this problem makes intuitive sense, and I've drawn out bounds, and I know obviously, that this statement will hold true, and the proof probably involves the intermediate theorem law, but I am just not sure how to construct my proof for it. Can I just say that for f(0)<x<f(1) then x=f(x) for some x (and repeat the same thing for my other two cases, when f(1)>f(0) and f(1)=f(0))?

Any help would be great

## Homework Statement

I have to prove that if a function is bounded for x between (0,1) and f(x) is bounded also between (0,1) and f(x) is continuos that there exists some x such that f(x)=x

## Homework Equations

Intermediate value theorem?

## The Attempt at a Solution

I know that this problem makes intuitive sense, and I've drawn out bounds, and I know obviously, that this statement will hold true, and the proof probably involves the intermediate theorem law, but I am just not sure how to construct my proof for it. Can I just say that for f(0)<x<f(1) then x=f(x) for some x (and repeat the same thing for my other two cases, when f(1)>f(0) and f(1)=f(0))?

Any help would be great

First off, you need the conditions you've listed to hold on ##[0,1]##. A counterexample on ##(0,1)## would be ##f(x)=x^2##.

You are correct that the Intermediate Value Theorem might prove useful. Think about the function ##g(x)=f(x)-x##.

Okay, I considered doing that; would I try to now show that g(x)=0 for some x?

Okay, I considered doing that; would I try to now show that g(x)=0 for some x?

Yes.

Okay, I think I got it now, thanks!
I just considered two points; g(1) and g(0), and showed how between those points, there always exists a zero as g(1) can be between 0 and -1 and g(0) is between 0 and 1, so by the intermediate value theorem, there has to be a zero between these two points for any g(x) that satisfy the properties of f(x)

That sounds like the right idea. You'll definitely need to beef up the rigor a bit to make it a proper proof.

D H
Staff Emeritus