Intermediate value theorem: Difference between revisions

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The intermediate value theorem states the following:

Consider anthe closed interval <math>I = [a,b]</math> of real numbers <math>\R</math> and a continuous function <math>f \colon I \to \R</math>. Then

*''Version I.'' if <math>u</math> is a number between <math>f(a)</math> and <math>f(b)</math>, that is, <math display="block">\min(f(a),f(b))<u<\max(f(a),f(b)),</math> then there is a <math>c\in (a,b)</math> such that <math>f(c)=u</math>.

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A [[Darboux function]] is a real-valued function {{mvar|f}} that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values {{mvar|a}} and {{mvar|b}} in the ___domain of {{mvar|f}}, and any {{mvar|y}} between {{math|''f''(''a'')}} and {{math|''f''(''b'')}}, there is some {{mvar|c}} between {{mvar|a}} and {{mvar|b}} with {{math|1=''f''(''c'') = ''y''}}. The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.

As an example, take the function {{math|''f'' : [0,&thinsp;∞) → [−1,&thinsp;1]}} defined by {{math|1=''f''(''x'') = sin(1/''x'')}} for {{math|''x'' > 0}} and {{math|1=''f''(0) = 0}}. This function is not continuous at {{math|1=''x'' = 0}} because the [[limit of a function|limit]] of {{math|1=''f''(''x'')}} as {{mvar|x}} tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the [[Conway's base 13 function]].

In fact, [[Darboux's theorem (analysis)|Darboux's theorem]] states that all functions that result from the [[derivative|differentiation]] of some other function on some interval have the [[intermediate value property]] (even though they need not be continuous).

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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|<nowiki>*</nowiki>}})

* 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|<nowiki>**</nowiki>}})

In fact, connectedness is a [[topological property]] and {{EquationNote|*|(*)}} 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 continuous, real-valued [[Function of a real variable|functions of a real variable]], to continuous functions in general spaces.

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

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==See also==

* {{annotated link|[[Mean value theorem}}]]

* {{annotated link|[[Non-atomic measure}}]]

* {{annotated link|[[Hairy ball theorem}}]]

* {{annotated link|[[Sperner's lemma}}]]

==References==