# One to one function is monotone?

1. Dec 6, 2015

### HaLAA

1. The problem statement, all variables and given/known data
A one to one function f: ℝ→ℝ is monotone, True or False

2. Relevant equations

3. 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]

2. Dec 6, 2015

### andrewkirk

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. Dec 6, 2015

### HaLAA

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. Dec 6, 2015

### Staff: Mentor

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. Dec 6, 2015

### andrewkirk

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. Dec 6, 2015

### Staff: Mentor

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. Dec 6, 2015

### WWGD

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

8. Dec 6, 2015

### SammyS

Staff Emeritus
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.