Continuity 100 Let f be a continuous function on a metric space X. Let Z(f)(the zero set of f) be the set of points EX f (a) -0. Prove that Z(f) s closed Solution PROOF :- We know that f(Z(f)) = {0} because the set Z(f) is dened to be all the values for which f(x) = 0 and so the set {0} is closed. Since f is continuous we know that f-1({0}) is closed. Therefore Z(f) is closed in X..