Bolzano's Theorem

If a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point.

Bolzano (1817) proved the theorem (which effectively also proves the general case of intermediate value theorem) using techniques which were considered especially rigorous for his time, but which are regarded as nonrigorous in modern times (Grabiner 1983).

REFERENCES:

Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, p. 143, 1967.

Bolzano, B. "Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege." Prague, 1817. English translation in Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem." Hist. Math. 7, 156-185, 1980.

Grabiner, J. V. "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus." Amer. Math. Monthly 90, 185-194, 1983.