Intermediate value theorem: Difference between revisions

Revision as of 07:29, 9 September 2023

Revision as of 20:52, 11 September 2023

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=== General metric and topological spaces ===

The intermediate value theorem is closely linked to the [[topology|topological]] notion of [[Connectedness (topology)|connectedness]] and follows from the basic properties of connected sets in metric spaces and connected subsets of '''R''' in particular:

* If <math>X</math> and <math>Y</math> are [[metric space]]s, <math>f \colon X \to Y</math> is a continuous map, and <math>E \subset X</math> is a [[Connected space|connected]] subset, then <math>f(E)</math> is connected. ({{EquationRef|<mathnowiki>(*)</mathnowiki>}})

* A subset <math>E \subset \R</math> is connected if and only if it satisfies the following property: <math>x,y\in E,\ x < r < y \implies r \in E</math>. ({{EquationRef|<mathnowiki>(**)</mathnowiki>}})

In fact, connectedness is a [[topological property]] and <math>{{EquationNote|*|(*)</math>}} generalizes to [[topological space]]s: ''If <math>X</math> and <math>Y</math> are topological spaces, <math>f \colon X \to Y</math> is a continuous map, and <math>X</math> is a [[connected space]], then <math>f(X)</math> is connected.'' The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of real valued functions of a real variable, to continuous functions in general spaces.

Recall the first version of the intermediate value theorem, stated previously:

{{math theorem|name=Intermediate value theorem|note=''Version I''|math_statement=Consider a closed interval <math>I = [a,b]</math> in the real numbers <math>\R</math> and a continuous function <math>f\colon I\to\R</math>. Then, if <math> u</math> is a real number such that <math>\min(f(a),f(b))< u < \max(f(a),f(b))</math>, there exists <math>c \in (a,b)</math> such that <math>f(c) = u</math>.}}

The intermediate value theorem is an immediate consequence of these two properties of connectedness:<ref>{{Cite book| url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis| last=Rudin|first=Walter| publisher=McGraw-Hill|year=1976|isbn=978-0-07-054235-8|___location=New York|pages=42, 93}}</ref>

{{math proof|proof= By <math>{{EquationNote|**|(**)</math>}}, <math>I = [a,b]</math> is a connected set. It follows from <math>{{EquationNote|*|(*)</math>}} that the image, <math>f(I)</math>, is also connected. For convenience, assume that <math>f(a) < f(b)</math>. Then once more invoking <math>{{EquationNote|**|(**)</math>}}, <math>f(a) < u < f(b)</math> implies that <math>u \in f(I)</math>, or <math>f(c) = u</math> for some <math>c\in I</math>. Since <math>u\neq f(a), f(b)</math>, <math>c\in(a,b)</math> must actually hold, and the desired conclusion follows. The same argument applies if <math>f(b) < f(a)</math>, so we are done. [[Q.E.D.]]}}

The intermediate value theorem generalizes in a natural way: Suppose that {{mvar|X}} is a connected topological space and {{math|(''Y'', <)}} is a [[total order|totally ordered]] set equipped with the [[order topology]], and let {{math|''f'' : ''X'' → ''Y''}} be a continuous map. If {{mvar|a}} and {{mvar|b}} are two points in {{mvar|X}} and {{mvar|u}} is a point in {{mvar|Y}} lying between {{math|''f''(''a'')}} and {{math|''f''(''b'')}} with respect to {{math|<}}, then there exists {{mvar|c}} in {{mvar|X}} such that {{math|1=''f''(''c'') = ''u''}}. The original theorem is recovered by noting that {{math|'''R'''}} is connected and that its natural [[Topological space|topology]] is the order topology.