The zero theorem

A particular case of the intermediate value theorem is among the most useful in practice: if a continuous function changes sign, it must vanish somewhere.

Theorem — The zero theorem

If $f$ is continuous on $[a,b]$ and $f(a)\cdot f(b) < 0$ (that is, $f(a)$ and $f(b)$ have opposite signs), then there exists at least one $c\in(a,b)$ with $f(c)=0.$

The zero theorem (due to Bolzano) is the foundation of the numerical methods for solving equations: it guarantees that a solution exists in a given interval, opening the way to the algorithms that locate it by successive approximations. It does not say, however, where the zero lies: only that there is one.

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Topics: Continuita
Concepts: Continuita · Teorema degli zeri · Teorema dei valori intermedi
Skills: Dimostrare
People: Bernard Bolzano