Rudin POMA: chapter 4 problem 14

  • #1
7
0

Homework Statement


Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##.

Homework Equations


Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##.

The Attempt at a Solution


Proof: let ##M = sup\ f(I)## and ##m = inf\ f(I)##. Assume that ##M ≠ m## and that ##f(x) ≠ x## for all ##x∈ I##.Then ##f## is strictly monotonic and so let ##L_1## = ##[f(m),f(M)]## and define ##L_n ##=## [f(first(L_{n-1}), f(second(L_{n-1}))]## (##n ≥ 2##). Then V=## \bigcap L_n## is non empty. Now let ##y∈V##. Then there exists ##x_1∈ V## such that ##f(x_1) = y##. Then if ##x_1 ≠y##, we have ##f(x_1) ≠ y##, which is a contradiction and so ##x_1 = y##. But this is contradicting our assumption. For the case where ##M = m## it is trivial as the function would be a constant one.
 

Answers and Replies

  • #2
tnich
Homework Helper
1,048
336

Homework Statement


Proof: let ##M = sup\ f(I)## and ##m = inf\ f(I)##. Assume that ##M ≠ m## and that ##f(x) ≠ x## for all ##x∈ I##.Then ##f## is strictly monotonic
I don't see how ##M>m## and ##f(x) ≠ x## imply that f(x) is monotonic.
 
  • #3
7
0
I don't see how ##M>m## and ##f(x) ≠ x## imply that f(x) is monotonic.
Since then ##f## would be a continous injective function on ##I## and so it is strictly monotonic.
 
  • #4
tnich
Homework Helper
1,048
336
Since then ##f## would be a continous injective function on ##I## and so it is strictly monotonic.
I agree that a continuous injective function is monotonic, but why is the function injective? How does ##f(x) \neq x## imply this?
 
  • #5
7
0
I agree that a continuous injective function is monotonic, but why is the function injective? How does ##f(x) \neq x## imply this?
Let ##x_1## ≠ ##x_2##. Then one gets ##f(x_1) ≠ f(x_2)## according to our assumption and so ##f## is injective.
 
  • #6
tnich
Homework Helper
1,048
336
Let ##x_1## ≠ ##x_2##. Then one gets ##f(x_1) ≠ f(x_2)## according to our assumption and so ##f## is injective.
Your argument is ##x_1 \neq f(x_1)## and ##x_2 \neq f(x_2)## implies ##f(x_1) \neq f(x_2)##?
 
  • #7
7
0
Your argument is ##x_1 \neq f(x_1)## and ##x_2 \neq f(x_2)## implies ##f(x_1) \neq f(x_2)##?
Yup.
 
  • #8
tnich
Homework Helper
1,048
336
Yup.
So ##f(x_1) \neq x_1 \neq x_2 \neq f(x_2)## implies ##f(x_1) \neq f(x_2)##? But ##\neq## is not transitive
 
Last edited by a moderator:
  • #9
34,172
5,786
Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##.
Basically, you're trying to show that every function f that maps the interval [0, 1] into itself must cross through or touch the line y = x at least once. I agree with @tnich that you are asserting that f is injective without showing why this must be true. In fact, for the given conditions, it doesn't have to be true (that f is injective).

In the graph below, f (dark curve) maps [0, 1] into [0, 1]. f is continuous on [0, 1], but neither injective nor surjective.

graph.png
 

Attachments

  • #10
tnich
Homework Helper
1,048
336
What could you do with the function ##g(x) \equiv f(x) - x##?
 
  • #11
7
0
So ##f(x_1) \neq x_1 \neq x_2 \neq f(x_2)## implies ##f(x_1) \neq f(x_2)##? But ##\neq## is not transitive
Yes you are right. My arguement is flawed in that step, but what if we delete that part from the proof? wouldn't the proof still hold?
 
  • #12
tnich
Homework Helper
1,048
336
Yes you are right. My arguement is flawed in that step, but what if we delete that part from the proof? wouldn't the proof still hold?
Can you make your sequence of intervals converge without it? Could you make your sequence converge even with it? Consider the case ##f(x) = 1-x##.
 
  • #13
7
0
Can you make your sequence of intervals converge without it? Could you make your sequence converge even with it? Consider the case ##f(x) = 1-x##.
Ah I see. Yes the arguement would not hold. Thanks for your help.
 
  • #14
7
0
What could you do with the function ##g(x) \equiv f(x) - x##?
I have solved the problem with this specific way, but I wanted to solve the problem with another approach.
 

Related Threads on Rudin POMA: chapter 4 problem 14

  • Last Post
Replies
1
Views
12K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
946
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
12
Views
7K
  • Last Post
Replies
2
Views
5K
Replies
1
Views
3K
  • Last Post
Replies
8
Views
3K
Replies
0
Views
1K
Top