PlanetMath: Bolzano's theorem

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Bolzano's theorem

(Theorem)

A continuous function can not change its sign without going through the zero.

This contents of Bolzano's theorem may be formulated more precisely as the

Theorem 1 If a real function $f$ is continuous on a closed interval $I$ and the values of $f$ in the end points of $I$ have opposite signs, then there exists a zero of this function inside the interval.

The theorem is used when using the interval halving method for getting an approximate value of a root of an equation of the form $f(x) = 0$ .

"Bolzano's theorem" is owned by pahio.

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Attachments:: proof of Bolzano's theorem (Proof) by cvalente
antipodal isothermic points (Application) by pahio

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Cross-references: root of an equation, interval halving, theorem, interval, function, end points, closed interval, real function, continuous function
There are 4 references to this entry.

This is version 2 of Bolzano's theorem, born on 2006-02-02, modified 2006-02-02.
Object id is 7584, canonical name is BolzanosTheorem.
Accessed 5102 times total.

Classification:

26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda

None.

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