One to one function is monotone?

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.

 

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).

 

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

 

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.