Let A be an open subset of and let be an increasing function. Prove that if f(A) is an open subset of then f is continuous.
Set , we are told that are open subsets of .
Let , so but is open so there exists so that . From here and on we will prove that the definition of continuity is satisfied for all . Since it means there exists so that . Now if and so . Certainly, , so the neighborhood of gets mapped within of . Thus, is continous at any point .