Bolzano Theorem (BT)

Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such that f(a) and f(b) are of opposite signs. Then there exists a number x₀[a, b] with f(x₀)=0.

Intermediate Value Theorem (IVT)

Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such that f(a)<f(b). Then for every y₀ such that f(a)<y₀<f(b) there exists a number x₀[a, b] with f(x₀)=y₀.

Clearly BT is only a special case of IVT. However it's interesting that the more general IVT could be deduced from its special case, BT. Indeed, let f(a)<y₀<f(b). Introduce a new function g(x)=f(x)-y₀. Then it follows that g(a)=f(a)-y₀<0 whereas g(b)=f(b)-y₀>0. Therefore, g(a) and g(b) are of opposite signs. Additionally, g is continuous wherever f is. In particular, g is continuous on [a, b] and thus satisfies the conditions of BT. Therefore, there exists x₀[a, b] such that g(x₀)=0. Written explicitly this says f(x₀)-y₀=0 or, finally,

Q.E.D.

A complete proof of the Bolzano Theorem appeares elsewhere.

Copyright © 1996-2009 Alexander Bogomolny