If
is continuous on a closed interval , and is any number between and inclusive, then there is at least one number in the closed interval
such that .
The theorem is proven by observing that is connected because the image of a connected set under
a continuous function is connected, where
denotes the image of the interval
under the function . Since is between and , it must be in this connected
set.
The intermediate value theorem (or rather, the space case with , corresponding to Bolzano's
theorem) was first proved by Bolzano (1817). While Bolzano's used techniques
which were considered especially rigorous for his time, they are regarded as nonrigorous
in modern times (Grabiner 1983).
Anton, H. Calculus with Analytic Geometry, 2nd ed. New York: Wiley, p. 189, 1984.Apostol,
T. M. "The Intermediate-Value Theorem for Continuous Functions." §3.10
in Calculus,
2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra.
Waltham, MA: Blaisdell, pp. 144-145, 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.Cauchy,
A. Cours d'analyse. Reprinted in Oeuvres, series 2, vol. 3, pp. 378-380.
English translation in Grabiner, J. V. The Origins of Cauchy's Rigorous Calculus.
Cambridge, MA: MIT Press, pp. 167-168, 1981.Grabiner, J. V.
"Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus."
Amer. Math. Monthly90, 185-194, 1983.