intermediate value theorem for extended real numbers

Theorem 1.

Let R¯ be the extended real numbers, and suppose f:R¯R¯ is a continuous functionMathworldPlanetmathPlanetmath. Suppose x1<x2R¯ are such that f(x1)f(x2). If y(f(x1),f(x2)), then for some c(x1,x2) we have


As ¯ is homeomorphic to [0,1], we can assume that f is a function f:[0,1]¯. For simplicity, let us also assume that x1=0,x2=1, and f(0)<f(1). Then for some ε>0 we have


Let g:[0,1] be the continuous function


Now g(0)=y-ε and g(1)=y+ε, so for some c(0,1), we have g(c)=y, and thus f(c)=y. ∎

Title intermediate value theorem for extended real numbers
Canonical name IntermediateValueTheoremForExtendedRealNumbers
Date of creation 2013-03-22 15:35:15
Last modified on 2013-03-22 15:35:15
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Theorem
Classification msc 26A06
Related topic ExtendedRealNumbers