zero of a function

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closed set

Suppose $X$ is a set and $f$ a complex (http://planetmath.org/Complex)-valued function $f\\colon X\\to\\mathbb{C}$ . 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 $\\mathbb{R}$ or $\\mathbb{C}$ 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 $z\\in\\mathbb{C}$ , define $\\hat{z}:X\\to\\mathbb{C}$ by $\\hat{z}(x)=z$ . Then $Z(\\hat{0})=X$ and $Z(\\hat{z})=\\varnothing$ if $z\eq 0$ .
•
Suppose $p$ is a polynomial (http://planetmath.org/Polynomial) $p\\colon\\mathbb{C}\\to\\mathbb{C}$ of degree $n\\geq 1$ . Then $p$ has at most $n$ zeroes. That is, $|Z(p)|\\leq n$ .
•
If $f$ and $g$ are functions $f\\colon X\\to\\mathbb{C}$ and $g\\colon X\\to\\mathbb{C}$ , then
$\\displaystyle Z(fg)$ $\\displaystyle=$ $\\displaystyle Z(f)\\cup Z(g),$
$\\displaystyle Z(fg)$ $\\displaystyle\\supseteq$ $\\displaystyle Z(f),$
where $f g$ is the function $x\\mapsto f(x)g(x)$ .
•
For any $f\\colon X\\to\\mathbb{R}$ , 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 $f:X\\to\\mathbb{C}$ is a function, then the support (http://planetmath.org/SupportOfFunction) of $f$ is given by:
$\\operatorname{supp}f=\\overline{Z(f)^{\\complement}}$
Further, if $f$ is continuous, then $Z(f)$ is closed (http://planetmath.org/ClosedSet) in $X$ (assuming that $\\mathbb{C}$ is given the usual topology of the complex plane where $\\{0\\}$ is a closed set).