# One to one function is monotone?

HaLAA

## Homework Statement

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

## 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]

Homework Helper
Gold Member
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.

HaLAA
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?

Mentor
2022 Award
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.

Homework Helper
Gold Member
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).

Mentor
2022 Award
(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.)

Gold Member
It is also an example of a function which is continuous when restricted to 2 dense subsets, but overall not continuous.

Staff Emeritus