least and greatest zero

Theorem.  If a real function f is continuousMathworldPlanetmathPlanetmath on the interval[a,b]  and has zeroes on this interval, then f has a least zero and a greatest zero.

Proof.  If  f(a)=0  then the assertion concerning the least zero is true.  Let’s assume therefore, that  f(a)0. 

The set  A={x[a,b]f(x)=0}  is bounded from below since all numbers of A are greater than a.  Let the infimumMathworldPlanetmathPlanetmath (http://planetmath.org/InfimumAndSupremumForRealNumbers) of A be ξ.  Let us make the antithesis, that  f(ξ)0.  Then, by the continuity of f, there is a positive number δ such that


Chose a number x1 between ξ and ξ+δ; then  f(x1)0,  but this number x1 is not a lower bound of A.  Therefore there exists a member a1 of A which is less than x1 (ξ<a1<x1).  Now  |a1-ξ|<|x1-ξ|<δ,  whence this member of A ought to satisfy that  f(a1)=0.  This a contradictionMathworldPlanetmathPlanetmath.  Thus the antithesis is wrong, and  f(ξ)=0.

This that  ξA  and ξ is the least number of A.

Analogically one shows that the supremumMathworldPlanetmath of A is the greatest zero of f on the interval.

Title least and greatest zero
Canonical name LeastAndGreatestZero
Date of creation 2013-03-22 16:33:22
Last modified on 2013-03-22 16:33:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 26A15
Synonym zeroes of continuous function
Related topic ZeroesOfAnalyticFunctionsAreIsolated