Microscopic tangency in one variable

One important comment about the proof of the Extreme Value Theorem is this.  The simple fact that the standard part of every hyperreal x satisfying a≤x≤b is in the original real interval [a, b] is the form that topological compactness takes in Robinson's theory: A standard topological space is compact if and only if every point in its extension is near a standard point, that is, has a standard part and that standard part is in the original space.

Suppose f[x] and f^′[x] are real functions defined on the interval (a, b), if we know that for all hyperreal numbers x with a<x<b and a≉x≉b, f[x + δx] - f[x] = f ' [x] · δx + ε · δx with ε≈0 whenever δx≈0, then arguments like the proof of the simple equivalency of limits and infinitesimals above show that f^′[x] is a uniform limit of the difference quotient functions on compact subintervals [α, β] ⊂ (a, b).  More generally, we can show:

Theorem: Uniform Differentiability

Suppose f[x] and f^′[x] are real functions defined on the open real interval (a, b).  The following are equivalent definitions of, "The function f[x] is smooth with continuous derivative f^′[x] on (a, b)."[x] on the interval (a, b).”

(a)  Whenever a hyperreal x satisfies a<x<b and x is not infinitely near a or b, then an infinitesimal increment of the extended dependent variable is approximately linear on a scale of the change, that is, whenever δx ≈ 0

f[x + δx] −f[x] = f^′[x] · δx + ε · δx with ε≈0

(b)  For every compact subinterval [α, β] ⊂ (a, b), the real limit

Underscript[lim, Δx0] (f[x + Δx] - f[x])/Δx = f^′[x] uniformly for α≤x≤b[x] uniformly for α ≤ x ≤ β

(c)  For every pair of hyperreal x_1≈x_2 with a<st[x_i] = c<b ,(f[x_2] - f[x_1])/(x_2 - x_1) ≈f^′[c]

(d)  For every c in (a, b), the real double limit, Underscript[lim, x_1c, x_2c] (f[x_2] - f[x_1])/(x_2 - x_1) = f^′[c]

(e)  The traditional pointwise defined derivative D_xf = Underscript[lim, Δx0] (f[x + Δx] - f[x])/Δx is continuous on (a, b).

Proof

See p.35 and 36.

Continuity of the derivative follows rigorously from the argument of Section 1, approximating the increment f[x_1] - f[x_2] from both ends of the interval [x_1, x_2].  

f[x_2] - f[x_1] = f^′ [x_1] · (x_2 - x_1) + ε_1 · (x_2 - x_1)

f[x_1] - f[x_2] = f^′ [x_2] · (x_1 - x_2) + ε_2 · (x_1 - x_2)

Adding, we obtain

0 = ((f^′ [x_1] - f^′ [x_2]) + (ε_1 - ε_2)) · (x_2 - x_1), so (f^′ [x_1] - f^′ [x_2]) = (ε_2 - ε_1) ≈0.

It certainly is geometrically natural to treat both endpoints equally, but this is a "locally uniform" approximation in real-only terms because the x values are hyperreal.  Uniformity gives a non-infiinitesimal explanation why the intuitive proof of the Fundamental Theorem works.  We take up the infinitesimal explanation in the next section.

In his General Investigations of Curved Surfaces (original in draft of 1825, published in Latin 1827, English translation by Morehead & Hiltebeitel, Princeton, NJ, 1902 and reprinted later by Raven Press), Gauss begins as follows:

A curved surface is said to posess continuous curvature at one of its points A, if the directions of all stright lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing trhough A.  This plane is said to touch the surface at the point A.

In The Handbook of Mathematical Logic, Jon Barwise (editor), North Holland Studies in Logic, nr. 90, Amsterdam 1977, Chapter A6, we show that this can be interpreted as C^1-embedded if we apply the condition to all points in the natural extension of the surface.

Created by Mathematica  (September 22, 2004)