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).
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. Monthly90, 185-194, 1983.
