Constructing a Non-One-to-One Function on [-1,1] Using Rational Numbers

[K29]

This question is at the end of a section on the Intermediate Value Theorem in my Real Analysis notes:
Find a continuous function f: [-1,1]->R which is one-to-one when restricted to rational numbers in [-1,1] but is not one-to-one on the whole interval [-1,1]

I can't figure it out. I've thought about piecewise functions and uhm circles, but I don't see how this is even possible. Any ideas?

PS if you have any ideas for[0,1] that would also help

 

[micromass]

Hi K29!

Let's see what we can do here. Take two distinct irrational point a and b. We will make a function such that f(a)=f(b).

Of course, we can't make a constant line between the function, so the function has to make some sort of arc between f(a) and f(b). That is, we will have to make f such that it decreases first and then increases, but such that it's still is one-to-one on the rationals.

No, if you make the decreasing part behave like the line y=-x, and the increasing part something like y=\pi x, then this would be something, right?

 

[K29]

Thanks, that helped me get to the right answer