Continuity & Extreme Values

We follow the idea of the proof in Section 1 for a real function f[x] on a real interval [a, b].  Coding our proof in terms of real functions.

There is a real function x_M[h] so that for each natural number h the maximum of the values f[x] for x = a + k Δx, k = 1, 2, ⋯, h and Δx = (b - a)/h occurs at x_M[h].  We can express this in terms of real functions using a real function indicating whether a real number is a natural number,

I[x] =  0, if x ≠ 1, 2, 3, ⋯ 1, if x = 1, 2, 3, ⋯

The maximum of the partition can be described by

(a≤x≤b & I[h (x - a)/(b - a)] = 1) ⇒ f[x] ≤f[x_M[h]]

We want to extend this function to unlimited "hypernatural" numbers.  The greatest integer function Floor[x] satisfies, I[Floor[x]] = 1,  0≤x - Floor[x] ≤1. The unlimited number 1/δ, for δ≈0 gives an unlimited H = Floor[x] with I[H] = 1 and

(a≤x≤b & I[H (x - a)/(b - a)] = 1) ⇒ f[x] ≤f[x_M[H]]

When the natural extension of the indicator function satisfies I[k] = 1, we say that k is a hyperinteger.  (Every limited hyperinteger is an ordinary positive integer.  As you can show with these functions.)

There is a greatest partition point of any number in [a, b], P[h, x] = a + Floor[h (x - a)/(b - a)] (b - a)/h with a≤P[h, x] ≤b & I[h (P[h, x] - a)/(b - a)] = 1 and 0≤x - P[h, x] ≤1/h.  When we take the unlimited hypernatural number H we have x - P[H, x] ≤1/H≈0 and P[H, x]  a partition point in the sense that (a≤x≤b & I[H (P[H, x] - a)/(b - a)] = 1), so we have

f[P[H, x]] ≤f[x_M[H]]

Let r_M = st[x_M[H]], the standard part.  Since a≤x_M[H] ≤b, a≤r_M≤b.  Continuity of the function in the sense x_1≈x_2 ⇒ f[x_1] ≈f[x_2]  gives

f[x] ≈f[P[H, x]] ≤f[x_M[H]] ≈f[r_M], so f[x] ≤f[r_M] for any real x in [a, b].

For more details see p.50.

Created by Mathematica  (September 22, 2004)