Could someone please check my work and make sure I'm using the correct theorems

Please state all definitions and theorems that you will need: Theorem 3.4.7 (a) A set S is open iff S = intS . Equivalently, S is open iff every point in S is an interior point of S. (b) A set S is closed iff its complement R\\S is open. - -..w Theorem 3.5.5 (Heine-Borel) A subset S of R is compact iff S is closed and bounded. Let f: D - R be continuous. For each of the following, prove or give a counterexample. 1. If D is open, then f(D) is open. What happens to D under f ? counterexample: f(x) = |x| D = R which is open by Theorem 3.4.7(a) f(D) = [0, oo) which is neither open or closed by the definition of the image of D in IR and Theorem 3.4.7 So, continuous functions don't necessarily map open sets to open sets. 2. If D is not compact, then f(D) is not compact. What happens to D under f ? counterexample: f(x) = sin x D = ( - 27, 27) which is not closed and thus not compact by Theorem 3.4.7(b) and by The Heine-Borel Theorem 3.5.5. f(D) = [ - 1, 1] which is closed by Theorem 3.4.7(b) and bounded by the definition of boundedness and thus compact by the Heine Borel Theorem 3.5.5 and thus not not compact. So, continuous functions don't necessarily map non-compact sets to non-compact sets. 3. If D is infinite, then f(D) is infinite. What happens to D under f ? counterexample: f(ac) = 2 D = R which is infinite f(D) = {2} which is finite as it is a single point and thus not infinite So, continuous functions don't necessarily map infinite sets to infinite sets