Intermediate value theorem: Difference between revisions

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==Relation to completeness==

The theorem depends on, and is equivalent to, the [[completeness of the real numbers]]. The intermediate value theorem does not apply to the [[rational number]]s '''Q''' because gaps exist between rational numbers; [[irrational numbers]] fill those gaps. For example, the function <math>f(x) = x^2-2</math> for <math>x\in\Q</math> satisfies <math>f(0) = -20</math> and <math>f(2) = 24</math>. However, there is no rational number <math>x</math> such that <math>f(x)=02</math>, because <math>\sqrt 2</math> is an irrational number.

==Proof==