Suppose f:R->R is continuous and define the zero set of f by Z(f)={x: f(x)=0}. Prove that Z(f) is a closed set. Give an example of a discontinuous function whose zero set is not closed. Help, please!
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Since f is continuous, it the preimage of closed sets are closed sets. {0} being a closed subset of R, f^-1(0) is closed. Consider g(x)= 0, if x<0 and g(x)=1, if x>=0. then g^-1(0) is open.
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