Intermediate Value Theorem

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).

REFERENCES:

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. Monthly 90, 185-194, 1983.