One to one function is monotone?

  • Thread starter HaLAA
  • Start date
  • #1
HaLAA
85
0

Homework Statement


A one to one function f: ℝ→ℝ is monotone, True or False

Homework Equations




The Attempt at a Solution


I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]
 

Answers and Replies

  • #2
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
4,039
1,618
You have not defined the value of f(x) outside of I, so the example does not meet the requirements of the question, which include that the domain be all of ##\mathbb{R}##.

I think the statement is false, but a little more work is needed to produce a counterexample.
 
  • #3
HaLAA
85
0
You have not defined the value of f(x) outside of I, so the example does not meet the requirements of the question, which include that the domain be all of ##\mathbb{R}##.

I think the statement is false, but a little more work is needed to produce a counterexample.
Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?
 
  • #4
fresh_42
Mentor
Insights Author
2022 Award
17,645
18,332
Yes. I thought f(x)=const. would also do, but it depends on whether f has to be strictly monotone or not. Yours is better.
 
  • #5
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
4,039
1,618
Yes, that rational/irrational function is a good one.
By the way, it's possible to extend your function in the OP to a non-monotone, injective function that has the entirety of ##\mathbb{R}## as domain. But the rational/irrational one in post 2 is easier to specify (albeit harder to visualize).
 
  • #6
fresh_42
Mentor
Insights Author
2022 Award
17,645
18,332
(albeit harder to visualize).
Since both are dense, it's just a big X. And it leads to interesting philosophical questions: what one draws is always discrete for you put carbon atoms on the paper. (I apologize, if that remark should be regarded as improper.)
 
  • #7
WWGD
Science Advisor
Gold Member
6,291
8,172
It is also an example of a function which is continuous when restricted to 2 dense subsets, but overall not continuous.
 
  • #8
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,693
1,273
Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?
Of course you could have f(x) = x outside of the interval (-1, 1) and f(x) = -x on the interval (-1, 1) .

or even simpler :

f(1) = -1, f(-1) = 1, otherwise, f(x) = x.
 

Suggested for: One to one function is monotone?

Replies
28
Views
357
Replies
13
Views
415
Replies
12
Views
578
Replies
4
Views
342
Replies
8
Views
478
Replies
8
Views
934
Replies
8
Views
361
Replies
19
Views
717
Replies
2
Views
601
  • Last Post
Replies
2
Views
467
Top