Intermediate value theorem: Difference between revisions
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Suppose <math>X</math> is a [[connected space]] and <math>f:X \to \R</math> is a [[continuous map]], where <math>\R</math> is the [[real line]] with the usual Euclidean topology. Then, if there exist <math>x_1,x_2 \in X</math> with <math>f(x_1) < f(x_2)</math>, <math>f(X)</math> contains the closed interval <math>[f(x_1),f(x_2)]</math>. In other words, <math>f</math> takes all intermediate values between <math>f(x_1)</math> and <math>f(x_2)</math>. | Suppose <math>X</math> is a [[connected space]] and <math>f:X \to \R</math> is a [[continuous map]], where <math>\R</math> is the [[real line]] with the usual Euclidean topology. Then, if there exist <math>x_1,x_2 \in X</math> with <math>f(x_1) < f(x_2)</math>, <math>f(X)</math> contains the closed interval <math>[f(x_1),f(x_2)]</math>. In other words, <math>f</math> takes all intermediate values between <math>f(x_1)</math> and <math>f(x_2)</math>. | ||
==Related facts== | |||
* [[Extreme value theorem]] | |||
==Facts used== | |||
# [[uses::Connectedness is continuous image-closed]] | |||
Revision as of 06:36, 23 December 2009
Statement
Suppose is a connected space and is a continuous map, where is the real line with the usual Euclidean topology. Then, if there exist with , contains the closed interval . In other words, takes all intermediate values between and .