Prove that X is discrete iff every function f : X-->R is continuous

Prove or Disprove that the following are homeomorphic using only material covered in class notes up through March 8.
a) S¹= {(x,y) | x² + y ² =1} and {(x,y) | max(|x|,|y|) = 1}, both with the subspace topologies of R².
b) R with the standard topology and R_cf with the finite complement topology.
c) [1,2) and {0}U(1,2) with the subspace topologies of R.
d) Grad The Zariski topology on R² and the finite complement topology on R²