Can the Intermediate Value Theorem Solve a Fixed Point Problem?

[ArcanaNoir]

So I was hanging out in my professor's office on Friday, playing maths, and I asked, "Couldn't I have a small problem to work on over the weekend?" So he thinks for a minute, and then he says, "If $f$ is a continuous function such that $f: [0,1] \to [0,1]$, then there exists a fixed $x \in [0,1] : f(x)=x$. Prove this" (I'm in his elementary proofs class.) So I take it in for a brief moment (too brief) and I smile and say, "easy, it's one of those basic calc function theorems, they're all in my book...Rolle's? no...ah yes! Intermediate value theorem will do the trick! I just have to look up the definition to be sure, then it will be simple." And he looks at me and says, "I'd like to give you a hint but it would then be obvious." And I shoot back straight away, "Hint? What you have said is already more than sufficient! This will take five minutes. I got this."

I. Got. This.

I so don't got this.

I first had a twinge of doubt when I opened my calc book and under the intermediate value theorem where I expect to see a proof I see instead: "This proof is given in more advanced books on calculus." Uh oh. Okay, I think, this is a rather abridged book, I've completed the entire thing, I can probably do this. They just don't want to put it here because it's like, chapter 1. ... It's only a matter of minutes before I realize this isn't the theorem that I need at all. I discover that I need a "fixed point theorem". Such a thing is definitely not in my calc book. So here I am on Saturday morning thinking, "omg, this is going to be very hard for someone at my level." But I better do it, because I said it was a cakewalk!

Anyone else said something to a prof they later regretted?

 

[micromass]

I think this is a valuable lesson: never tell a professor that something was easy, especially when you did not prove it yet

Anyway:

Spoiler

Just look at the function $g(x)=f(x)-x$ and apply the intermediate value theorem.

 

[ArcanaNoir]

Thank you for the spoiler alert! I look forward to reading what I imagine is your proof when I am done with mine :)

 

[disregardthat]

Draw a picture!

 

[ArcanaNoir]

Draw a picture!

lol that's what my friend did and my professor said that was sort of philosophical and less than a proof.

 

[Hurkyl]

lol that's what my friend did and my professor said that was sort of philosophical and less than a proof.

You're supposed to turn the picture into a proof, not submit it as a proof.

 

[micromass]

Anyway, your professor is pretty smart. Because even if you could answer his question on the spot, he could immediately ask to prove the same thing for $g:[0,1]^2\rightarrow [0,1]^2$. Or, god forbid, for $g:[0,1]^n\rightarrow [0,1]^n$. These things are a bit... harder.

So, whatever you did, you would always have lost.

 

[BrianMath]

Would you mind if I post my attempt at a proof?

 

[ArcanaNoir]

I think I proved it (at least I've finished the piece of paper I'm going to hand him and hope he finds adequate), so anyone who wants to post a proof can feel free now.

 

[BrianMath]

I think I proved it (at least I've finished the piece of paper I'm going to hand him and hope he finds adequate), so anyone who wants to post a proof can feel free now.

Okay, here's my attempt. Hopefully it's correct. Tell me if it's similar to what you have.

 

[ArcanaNoir]

Okay, here's my attempt. Hopefully it's correct. Tell me if it's similar to what you have.

Kinda. I took cases, the most important being that if f(0) not 0 and f(1) not one then f(x)> x and f(x)< x and then since its continuous it has to "switch" somewhere and converges to a point where |f(x)-x| goes to zero so f(x)=x on account of the limit as it approaches from each side.

 

[qspeechc]

I also thought this would take me 5 minutes, in fact I remember working on this problem several years ago and making heavy weather of it! Well I was thinking about it while I was driving to the nursery and it took me about 20 minutes to get the proof and sort out all the details. It's basically the fact that f has a maximum and the intermediate value theorem, and looking at cases.

 

[xxChrisxx]

I go to the pub at weekends.