Continuity 100 Let f be a continuous function on a metric space X. L.pdf
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The document discusses a proof that the zero set of a continuous function f on a metric space is closed. It states that since the set of points where f(x) = 0 is defined to be the zero set, and the preimage of a closed set under a continuous function is also closed, the zero set must be closed. Thus, the argument concludes that z(f) is closed in the metric space.
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Continuity 100 Let f be a continuous function on a metric space X. L.pdf
- 1. 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.