Let f : R ? R be a function.
(a) Prove that f is continuous on R if and only if, for everyopen set U ? R, the preimage f ?1 (U) = {x ? R : f(x) ? U} isopen.
(b) Use part (a) to prove that if f is continuous on R, its zeroset Z(f) = {x ? R : f(x) = 0} is closed.
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