zero of a function

closed set

Suppose $X$ is a set and $f$ a complex (http://planetmath.org/Complex)-valued function  $f:X\to ℂ$.  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 $\widehat{z}:X\to ℂ$ by $\widehat{z}(x)=z$. Then $Z(\widehat{0})=X$ and $Z(\widehat{z})=\mathrm{\varnothing }$ if $z\ne 0$.

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

• If $f$ and $g$ are functions $f:X\to ℂ$ and $g:X\to ℂ$, then

  $Z(fg)$ $=$ $Z(f)\cup Z(g),$

  $Z(fg)$ $\supseteq$ $Z(f),$

  where $fg$ is the function  $x\mapsto f(x)g(x)$.

• For any $f:X\to ℝ$, 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 (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 $ℂ$ 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	2013-03-22 14:00:58
Last modified on	2013-03-22 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