Let $f:S\to\mathbb{R}$ a continuous function. Prove that $Z(f)=\{p\in S:f(p)=0\}$ is closed.

Question

I know that I have to solve this problem indirectly and using the fact that the inverse image of open set is open.

Theorem 4.3

Let $S$ and $T$ be metric spaces and $E $ a subset of $S$. A function $f : E \rightarrow T$ is continuous on $E$ if and only if $f^{−1}(O)$ is open in $E$ for every open set $O \subset T $.

I just don’t know how start!

Thanks.

Answer 1

As $f^{-1}(T \backslash O)=E \backslash f^{-1}(O)$ you can first prove that the inverse image of a closed set is closed.

Then you can prove that $\{0\}$ is closed to conclude.