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May 5th, 2007, 07:09 AM   #1
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Intermediate values theorem

Here's a spectacular, though very easy, application of the intermediate values theorem: prove that there exist two diametrically opposed points of the earth with equal temperature (assuming the temperature is continuously distributed, and assimilating the earth to a perfect 3 dimensional ball).
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May 18th, 2007, 11:22 AM   #2
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Let D(p) be the diametrically opposed point of p and let T(p) be the temperature of p.
Manifestly f(p)=T(p)-T(D(p)) is a continuous function defined on a compact set, and f(D(p))=-f(p) ,
so there is Po such that f(Po)=0 and finally T(Po)=T(D(Po)) (q.e.d)
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