If f from R to R is continuous and open, show that f is strictly monotone.
If f is not monotone, then there are two distinct points a<b such that f(a)=f(b). Then by Weierstrass' theorem f has a maximum C and minimum c on the closed interval [a,b]. We can assume that at least one of the numbers f(C) or f(c), say f(C) differs from f(a)=f(b).
Then the image open interval f( (a,b) ) = (f(a), C] is not open, which contradicts to openness of f.
I am extremely satisfied! It passed all tests on Dr Java and I also ran it through cmd junit. All 190 tests worked. I'm very impressed. I almost lost faith that there's anyone out there to help until I found your site.
I'm extremely satisfied. You have shown excellence.
I have good news to share. I'm starting master degree in computer science from August 2017.
I hope to depend on you for the next 2 years and I hope you can help me on thesis also.
You are the only reliable help I have.
Thanks a lot!!!!