(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If H:I[tex]\rightarrow[/tex]I is a monotone and continuous function, prove that H is a homeomorphism if either

a) H(0) = 0 and H(1) = 1

or b) H(0) = 1 and H(1) = 0.

2. Relevant equations

3. The attempt at a solution

So if I can prove H is a homeomorphism for a), b) follows from the fact that the map defined by t[tex]\rightarrow[/tex]H(1-t) is also a homeomorphism because it is the composite of two homeomorphisms.

H is obviously one-to-one, but I don't know how to "show" this.

At first I figured that I should assume |H(t_{1})-H(t_{2})| < [tex]\delta[/tex] for some [tex]\delta[/tex] > 0 and try to show that there exists a [tex]\epsilon[/tex] > 0 such that |t_{1}- t_{2}| < [tex]\epsilon[/tex]. I didn't know where to go after this, so I tried the sequence definition for continuity and I got nowhere.

Any suggestions?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Monotonic and Continuous function is homeomorphism

**Physics Forums | Science Articles, Homework Help, Discussion**