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Intermediate Value Theorem


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The following theorem stating an important property of continuous functions:

Theorem

If the real function f is continuous on the closed interval [a, b] and η is a real number between f(a) and f(b), then, for some c in (a, b), f(c)=η.

The theorem is useful for locating roots of equations. For example, suppose that f(x)=x−cos x. Then f is continuous on [0, 1], and f(0)<0 and f(1)>0, so it follows from the Intermediate Value Theorem that the equation f(x)=0 has a root in the interval (0, 1).

Subjects: Mathematics.


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