One to one function is monotone?

[HaLAA]

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]

 

[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.

 

[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?

 

[fresh_42]

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.

 

[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).

 

[fresh_42]

(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.)

 

[WWGD]

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

 

[SammyS]

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.