zero of a function


closed setPlanetmathPlanetmath

Suppose X is a set and f a complex ( functionf:X.  Then a zero of f is an elementxX  such that  f(x)=0.  It is also said that f vanishes at x.

The zero set of f is the set


Remark. When X is a “simple” space, such as or a zero is also called a root.  However, in pure mathematics and especially if Z(f) is infiniteMathworldPlanetmathPlanetmath, it seems to be customary to talk of zeroes and the zero set instead of roots.


  • For any z, define z^:X by z^(x)=z. Then Z(0^)=X and Z(z^)= if z0.

  • Suppose p is a polynomialPlanetmathPlanetmath (  p:  of degree n1.  Then p has at most n zeroes. That is, |Z(p)|n.

  • If f and g are functions f:X and g:X, then

    Z(fg) = Z(f)Z(g),
    Z(fg) Z(f),

    where fg is the function  xf(x)g(x).

  • For any f:X, then


    where fn is the defined fn(x)=(f(x))n.

  • If f and g are both real-valued functions, then

  • If X is a topological spaceMathworldPlanetmath and f:X is a function, then the supportMathworldPlanetmath ( of f is given by:


    Further, if f is continuousMathworldPlanetmathPlanetmath, then Z(f) is closed ( in X (assuming that 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