Bolzano's theorem

Bolzano's theorem states that a continuous real-valued function with opposite signs at two points has at least one zero between them. It is a direct consequence of the intermediate value property.

Intermediate value theorem: A continuous function on an interval takes every value between its endpoint values. Bolzano's theorem is the zero-value case of this broader result.
Real number: A number represented by a point on the continuous number line. The theorem concerns real-valued functions on real intervals.
Root-finding algorithm: A computational procedure for approximating solutions of an equation. Bolzano's theorem certifies that a bracketed root-finding problem has a solution.
Darboux's theorem: A derivative takes every intermediate value between any two values it attains, even when it is not continuous. It has an intermediate-value conclusion without assuming the function itself is continuous.
Bisection method: A root-finding algorithm that repeatedly halves an interval whose endpoint values have opposite signs. The theorem guarantees that every interval retained by bisection still contains a root.
Interval (mathematics): A subset of the real line containing every number between any two of its elements. The points between the two sign-changing inputs form the domain region where the zero is guaranteed.
Regula falsi: A root-finding method that estimates a root using a secant line while preserving a sign-changing interval. It uses the same guaranteed root bracket as Bolzano's theorem.
Discontinuous function: A function that fails to be continuous at one or more points of its domain. Without continuity, opposite endpoint signs need not force any zero between them.
Continuity: A function is continuous when small input changes produce arbitrarily small output changes. Continuity prevents the function from changing signs without taking the value zero.
Supremum: The least upper bound of a set of real numbers. A standard completeness-based proof locates a boundary point where the function must vanish.

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Bernard BolzanoBroader topic: Bolzano gave an early rigorous proof of this result using continuity and the intermediate value property.