Can the Intermediate Value Theorem Solve a Fixed Point Problem?

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A student sought a weekend problem from their professor, who posed a challenge involving a continuous function f: [0,1] → [0,1] and the existence of a fixed point x such that f(x) = x. Initially confident, the student quickly realized that the Intermediate Value Theorem was not the appropriate tool for the proof, as it required knowledge of a fixed point theorem not covered in their calculus book. This led to feelings of doubt about tackling the problem. The discussion highlights the lesson learned about overestimating one's understanding in academic settings. Other participants shared their experiences and attempts at proving the theorem, discussing strategies involving continuity and the behavior of functions at endpoints. The conversation emphasized the complexity of mathematical proofs and the importance of thorough preparation.
ArcanaNoir
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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?
 
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I think this is a valuable lesson: never tell a professor that something was easy, especially when you did not prove it yet :biggrin:

Anyway:


Just look at the function g(x)=f(x)-x and apply the intermediate value theorem.
:biggrin:
 
Thank you for the spoiler alert! I look forward to reading what I imagine is your proof when I am done with mine :)
 
Draw a picture!
 
disregardthat said:
Draw a picture!

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

So, whatever you did, you would always have lost. :-p
 
Would you mind if I post my attempt at a proof? :smile:
 
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.
 
  • #10
ArcanaNoir said:
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. :smile:
 

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  • #11
BrianMath said:
Okay, here's my attempt. Hopefully it's correct. Tell me if it's similar to what you have. :smile:

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.
 
  • #12
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.
 
  • #13
I go to the pub at weekends.
 

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