1. Let A=R. Then A is open and f(A) = (0,1].
2 is obviously false (take A to be a singleton). Did you mean f(A) is closed for closed sets A? If so, use the same example as (1).
Consider f(x) = 1/(1+x^2) as a function from R to R with the usual metric. Are the following statements true or false (give proofs or counterexamples):
1. f(A) is open for all open sets A in R
2. f(A) is open for all closed sets A in R