- #1

jjou

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**1. Let [tex]f:\mathbb{R}\rightarrow\mathbb{R}[/tex] 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: [tex]U\subset\mathbb{R}[/tex] is open iff [tex]f(U)\subset\mathbb{R}[/tex] is open.

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