Intermediate Value Theorem

The Intermediate Value Theorem: if a continuous function takes values of opposite signs at two points, it takes the value zero at some point between them.

A fundamental topological property of continuous functions that cannot be proved without a rigorous definition of continuity and the real number system. Bolzano gave the first rigorous proof (1817) and Cauchy gave another — the theorem is philosophically significant because it shows that intuitive geometric truths require the completeness of the real numbers.