In topology: homeomorphism v. monotone function

[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. :)

 

[morphism]

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

 

[HallsofIvy]

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

 

[jjou]

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