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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- Nov 13th 2012, 05:39 PMlovesmathClosed Sets
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! - Nov 14th 2012, 08:14 AMModusPonensRe: Closed Sets
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.