Continuity & Extreme Values

A foundation of real analysis is:

Theorem: The Extreme Value Theorem

Suppose a function f[x]  is continuous on a compact interval [a, b].  Then  f[x]  attains both a maximum and minimum, that is, there are points x_MAXCell[] and  x_min   in  [a, b], so that for every other  x  in  [a, b], f[x_min] ≤f[x] ≤f[x_MAX].

Formulating the meaning of "continuous" is a large part of making this result precise.  We will take the intuitive "definition" that f[x] is continuous means that if an input value x_1is close to another, x_2, then the output values are close.  We summarize this as: f[x] is continuous if and only if

a≤x_1≈x_2≤b ⇒ f[x_1] ≈f[x_2]

Given this property of f[x], if we partition [a, b] into tiny increments,

a<a + (1 (b - a))/H<a + (2 (b - a))/H<⋯<a + k(b - a)/H<⋯<b

the maximum of the finite partition occurs at one (or more) of the points x_M = a + k(b - a)/H.  This means that for any other partition point x_1 = a + j(b - a)/H, f[x_M] ≥f[x_1].

Any point a≤x≤b is within 1/H of a partition point x_1 = a + j(b - a)/H, so if H is very large, x≈x_1 and

f[x_M] ≥f[x_1] ≈f[x]

so we have found the approximate maximum.

It is not hard to make this idea into a sequential argument where x_M[H] depends on H, but there is quite some trouble to make the sequence x_M[H] converge (using some form of compactness of [a, b].)  Robinson's theory simply shows that the hyperreal x_M chosen when 1/H is infinitesimal, is infinitely near an ordinary real number where the maximum occurs. (A very general and simple re-formulation of compactness.)  We complete this proof as a simple example of Keisler's Axioms in Section 2.

Created by Mathematica  (September 22, 2004)