# Thread: Closed Sets

1. ## Closed 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!

2. ## Re: 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.