# In topology: homeomorphism v. monotone function

1. Feb 2, 2008

### jjou

1. Let $$f:\mathbb{R}\rightarrow\mathbb{R}$$ be a bijection. Prove that f is a homeomorphism iff f is a monotone function.

I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove this using contradiction.

Assume it is a homeomorphism but not monotone. Then there exists a,b,c in R s.t. a<b and a<c but f(a)<f(b) and f(a)>f(c). I think the statement I want to eventually contradict is the following: $$U\subset\mathbb{R}$$ is open iff $$f(U)\subset\mathbb{R}$$ is open.

Could someone give me a small hint? Thanks. :)

2. Feb 3, 2008

### morphism

Use the intermediate value theorem (draw a sketch!).

3. Feb 3, 2008

### HallsofIvy

Staff Emeritus
Can you prove that if f is not monotone, then f(x)= f(y) for some $x\ne y$?

4. Feb 3, 2008

### jjou

Thanks! I got it very quickly using both of your suggestions. :D