Exercise 4.3

Question

Exercise 3:
    Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the {\it zero set} of $f$) be the set of all $p\in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.

ghostofgarborg · Accepted Answer

Note that $\{ 0 \}$ is closed, so that $Z(f) = f^{-1}(\{ 0 \})$ is closed by the continuity of $f$.

Amit Awasthi · Answer

\begin{proof}
      Let $A$ be the set of all $f(p)$ with $p \in Z(f)$. $A$ only contains 0, since that is how we defined $Z(f)$ above. Now consider $A^c$. This set is open since it is the union of two open sets, i.e. $A^c = (-\infty,0) \cup (0,\infty)$. Since $f$ is continuous, the set $B$ which maps onto $A^c$ is open in $X$. Since $Z(f) = B^c$, we know that $Z(f)$ is closed.
    \end{proof}

Exercise 4.3

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Exercise 4.3

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