Let f be defined and continuous on a closed set S in R. Let A={x: x$\displaystyle \in$S and f(x)=0}.
Prove that A is a closed subset of R .
Hint: If $\displaystyle f$ is continues and $\displaystyle f(p)\not=0$ then there is an open interval such that $\displaystyle p\in (s,t)$ and $\displaystyle f$ is non-zero on $\displaystyle (s,t)$.
Hence, does this show the complement open?