zero of a function
Suppose $X$ is a set and $f$ a complex (http://planetmath.org/Complex)valued function $f:X\to \u2102$. 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 $\u2102$ 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

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For any $z\in \u2102$, define $\widehat{z}:X\to \u2102$ by $\widehat{z}(x)=z$. Then $Z(\widehat{0})=X$ and $Z(\widehat{z})=\mathrm{\varnothing}$ if $z\ne 0$.

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Suppose $p$ is a polynomial^{} (http://planetmath.org/Polynomial) $p:\u2102\to \u2102$ of degree $n\ge 1$. Then $p$ has at most $n$ zeroes. That is, $Z(p)\le n$.

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If $f$ and $g$ are functions $f:X\to \u2102$ and $g:X\to \u2102$, then
$Z(fg)$ $=$ $Z(f)\cup Z(g),$ $Z(fg)$ $\supseteq $ $Z(f),$ where $fg$ is the function $x\mapsto f(x)g(x)$.

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For any $f:X\to \mathbb{R}$, then
$$Z(f)=Z(f)=Z({f}^{n}),$$ where ${f}^{n}$ is the defined ${f}^{n}(x)={(f(x))}^{n}$.

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If $f$ and $g$ are both realvalued functions, then
$$Z(f)\cap Z(g)=Z({f}^{2}+{g}^{2})=Z(f+g).$$ 
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If $X$ is a topological space^{} and $f:X\to \u2102$ is a function, then the support^{} (http://planetmath.org/SupportOfFunction) of $f$ is given by:
$$\mathrm{supp}f=\overline{Z{(f)}^{\mathrm{\complement}}}$$ Further, if $f$ is continuous^{}, then $Z(f)$ is closed (http://planetmath.org/ClosedSet) in $X$ (assuming that $\u2102$ is given the usual topology of the complex plane where $\{0\}$ is a closed set).
Title  zero of a function 

Canonical name  ZeroOfAFunction 
Date of creation  20130322 14:00:58 
Last modified on  20130322 14:00:58 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  30 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 26E99 
Synonym  zero 
Synonym  vanish 
Synonym  vanishes 
Related topic  SupportOfFunction 
Defines  zero set 