zero of a function

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 $z\neq 0$.

• 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 $fg$ 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).

Title zero of a function ZeroOfAFunction 2013-03-22 14:00:58 2013-03-22 14:00:58 mathcam (2727) mathcam (2727) 30 mathcam (2727) Definition msc 26E99 zero vanish vanishes SupportOfFunction zero set