intermediate value theorem


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intermediate value theorem

[‚in·tər′mēd·ē·ət ¦val·yü ′thir·əm]
(mathematics)
If ƒ(x) is a continuous real-valued function on the closed interval from a to b, then, for any y between the least upper bound and the greatest lower bound of the values of ƒ, there is an x between a and b with ƒ(x) = y.
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By Intermediate value theorem there is a fixed point between [x.
By the Intermediate value theorem, there is a fixed point between [x.
j]) thus by the intermediate value theorem f admits a fixed point in the interval [[x.
by Intermediate Value theorem, there is a point 'x' such that g(x)=0 that is f(x) =x.
Since there is no intermediate value theorem for complex valued functions, the proof does not carry over to the case B [subset] C, though B may be regarded as two dimensional in this case.
Here, as in the case of the Intermediate Value Theorem, there are constructive substitutes, but they are essentially weaker, in this case significantly so.
Thus, in the basic example of the Intermediate Value Theorem, the classicist proves that intermediate values are taken on by any continuous function on a compact set, whereas the constructivist proves this only for restricted subclasses of functions, e.
5 For conciseness, we are bypassing applications of the Intermediate Value Theorem.
Bernard Bolzano proved the intermediate value theorem.
Most calculus and analysis texts contain a proof of the intermediate value theorem, and often they have a few casual comments about its significance.
When this attitude is brought to bear on the intermediate value theorem, it is perfectly natural to conclude that, until Bolzano, we couldn't really be sure the theorem is true.
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