Could someone please check my work

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(cc) = |w| D = IR which is open f (D) = [0, 00) which is neither open or closed 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(m) = sinx D =2 ( 27f, 27r) which is not closed and thus not compact. f(D) = [ - 1, 1] which is closed and bounded 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(az) = 2 D = R which is infinite . f(D) = {2} So, continuous functions don't necessarily map infinite sets to infinite sets