Suppose $X$ is a set and $f$ a complex-valued function $f\colon X\to \C$ Then a zero of $f$ is an element $x\in X$ , such that $f(x) = 0$ It is also said that $f$ vanishes at $x$

The zero set of $f$ is the set $$Z(f) := \{ x\in X \mid f(x)=0\}.$$

Remark. When $X$ is a ``simple'' space, such as $\R$ or $\C$ a zero is also called a root. However, in pure mathematics and especially if $Z(f)$ is infinite, it seems to be customary to talk of zeroes and the zero set instead of roots.

Examples

• For any $z\in \C$ define $\hat{z}:X\to \C$ by $\hat{z}(x)=z$ Then $Z(\hat{0})=X$ and $Z(\hat{z})=\varnothing$ if $z\ne 0$
• Suppose $p$ is a polynomial $p\colon\C\to\C$ , of degree $n\ge 1$ Then $p$ has at most $n$ zeroes. That is, $|Z(p)|\le n$
• If $f$ and $g$ are functions $f\colon X\to\C$ and $g\colon X\to\C$ then \begin{eqnarray*} Z(fg)&=&Z(f)\cup Z(g),\\ Z(fg)&\supseteq& Z(f), \end{eqnarray*}where $fg$ is the function $x\mapsto f(x) g(x)$
• For any $f\colon X\to \R$ then $$Z(f)=Z(|f|)=Z(f^n),$$ where $f^n$ is the defined $f^n(x)=(f(x))^n$
• If $f$ and $g$ are both real-valued functions, then $$Z(f)\cap Z(g)=Z(f^2+g^2)=Z(|f|+|g|).$$
• If $X$ is a topological space and $f:X\to \C$ is a function, then the support of $f$ is given by: $$\operatorname{supp} f = \overline{Z(f)^\complement}$$ Further, if $f$ is continuous, then $Z(f)$ is closed in $X$ (assuming that $\C$ is given the usual topology of the complex plane where $\{0\}$ is a closed set).