Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.