SOME NOTES ON NON-STANDARD DIFFERENTIATION AND INTEGRATION

TRISTRAM DE PIRO

Abstract. This short paper provides some basic results required to define non-standard differentiation and integration.

We assemble here some basic results concerning non-standard differentiation and integration. We will assume some familiarity with ideas in mathematical logic and properties of the real numbers R. A far more detailed account can be found in [2].

Lemma 0.1. There exists a structure (R^{∗}, +, ., <), that is an elementary extension of the real numbers R, with the property that it contains an element ϵ, such that;

0 < ϵ < r, for every r ∈ R with 0 < r (†)

Proof. The proof is a straightforward application of the compactness theorem from first order logic. One considers the following theory in the language L_{R} ∪ {c};

T h(R, +, ., <) ∪ {0 < c < r : r ∈ R}

Any finite set S of sentences from this theory is satisfied by the structure (R, +, ., <) itself, as one can interpret the constant c to be any real number r′, such that;

0 < r′ < min_{1≤i≤n}r_{i}

where r_{i} ∈ R are the finitely many constant terms appearing in the set S. Hence, the theory has a model (R^{∗}, +, ., <) with the desired properties. □

Thanks to my colleague Professor Carlo Toffalori, at The University of Camerino, for a helpful discussion and the use of his office from 14th-16th May, 2008.

Definition 0.2. We will refer to any element ϵ ∈ R^{∗}, with the property (†), as a positive infinitesimal, and, any element r′ ∈ R^{∗}, with the property that there exist {r_{1}, r_{2}} ⊂ R, such that r_{1} < r′ < r_{2}, as bounded. We refer to elements in R^{∗}, but not in R, as non-standard. We refer to any structure R^{∗} satisfying Lemma 0.1, as a non-standard extension of R.

We then have the following;

Lemma 0.3. For every bounded r′ ∈ R^{∗}, there exists a unique r ∈ R, such that;

r′ = r + ϵ or r′ = r − ϵ or r′ = r

where ϵ is a positive infinitesimal. Proof. As r′ is bounded, the sets;

S_{1} = {r ∈ R : r ≤ r′}, S_{2} = {r ∈ R : r′ ≤ r}

are both non-empty. By straightforward properties of R, there exist {s_{1}, s_{2}} ⊂ R, such that;

s_{1} = sup(S_{1}) and s_{2} = inf(S_{2})

It is easy to see that s_{1} = s_{2}, using the fact that R^{∗} satisfies the first-order property of R;

∀x∀y(x < y → ∃z(x < z < y))

Let r = s_{1} = s_{2}, then clearly r must belong to either S_{1} or S_{2}, the existence result then follows from considering the alternatives r ∈ S_{1}∩S_{2}^{c}, r ∈ S_{1}^{c} ∩ S_{2} or r ∈ S_{1} ∩ S_{2}. For uniqueness, suppose that there existed distinct elements {r_{1}, r_{2}} ⊂ R, and, without loss of generality, positive infinitesimals {ϵ_{1}, ϵ_{2}}, such that r = r_{1} + ϵ_{1} = r_{2} + ϵ_{2}. Then;

(r_{1} − r_{2}) = (ϵ_{2} − ϵ_{1})

However, it is straightforward to show that if ϵ_{2} −ϵ_{1} ≠ 0, there exists a positive infinitesimal c, such that ϵ_{2} − ϵ_{1} = c or ϵ_{2} − ϵ_{1} = −c. In any case, this contradicts the fact that r_{1} − r_{2} ≠ 0.

□

Definition 0.4. Given a bounded r′ ∈ R^{∗}, we refer to the r provided by the previous lemma as the standard part st(r′) of r′. Given r ∈ R, we refer to the set;

V_{r} = {r′ ∈ R^{∗} : st(r′) = r}

as an infinitesimal neighborhood of r.

We then have;

Lemma 0.5. The infinitesimal neighborhoods V_{r}, for r ∈ R, define a partition of the bounded elements of R^{∗}. Moreover, each infinitesimal neighborhood V_{r} contains non-standard elements, that is elements distinct from r.

Proof. If r_{1} ≠ r_{2} belong to R, then the infinitesimal neighborhoods V_{r_{1}} and V_{r_{2}} are disjoint, by the uniqueness part of Lemma 0.3. It follows easily that the infinitesimal neighborhoods define a partition of the bounded elements of R. If ϵ is a positive infinitesimal, then r + ϵ is a non-standard element, belonging to the infinitesimal neighborhood V_{r} of r. □

We then consider how to define real functions on a non-standard extension R^{∗};

Lemma 0.6. Let f(x) be a real function on the bounded interval (a, b) ⊂ R, then f extends to a function on the bounded interval (a, b) ⊂ R^{∗}, for a non-standard extension of R

Proof. One should go through the proof of Lemma 0.1, replacing the structure (R, +, ., <) with (R, +, ., <, f). The result then follows from the fact that the statement;

∀x[(a < x < b) → ∃!y(f(x) = y)]

is elementary for the theory T h(R, +, ., <, f).

□

With this basic set up, we can then define non-standard versions of continuity and differentiation;

Definition 0.7. Non-Standard Continuity

Let f(x) be a real function on the bounded interval (a, b) ⊂ R, and consider its extension to R^{∗}. We say that f is non-standard continuous at x ∈ (a, b), for x ∈ R, if;

f(x + ϵ) ∈ V_{f(x)}, for any ϵ ∈ V_{0}

We then have;

Lemma 0.8. f is non-standard continuous at x ∈ (a, b) iff it is continuous at x ∈ (a, b).

Proof. The proof is an elementary exercise, using the method of Lemma 0.3 and the usual definition of continuity in terms of limits. □

Definition 0.9. Non-Standard Differentiation

Let hypothesis be as in the previous definition. Then we say that f is non-standard differentiable, at x ∈ (a, b), if there exists c ∈ R, such that;

\frac{f(x+ϵ)−f(x)}{ϵ} ∈ V_{c}, for every infinitesimal ϵ ∈ V_{0}.

We then define the non-standard derivative of f at x to be c.

Lemma 0.10. f is non-standard differentiable at x ∈ (a, b) iff it is differentiable at x ∈ (a, b). Moreover, in this case, the non-standard derivative equals the usual derivative.

Proof. Again, the proof is an easy consequence of the usual definition of differentiability at x ∈ (a, b) in terms of limits, that;

lim_{h→0} \frac{f(x+h)−f(x)}{h}

exists, and arguments, using the method of Lemma 0.3.

□

We now consider the non-standard approach to integration;

Definition 0.11. Let f(x) be a continuous function on the bounded interval [a, b]. For a real number c, with 0 < c < (b − a), we define the Riemann sum R_{f}(c) to be; R_{f}(c) = ∑_{j=0}^{N(c)} f(a + jc)c

where N(c) is the greatest positive integer n such that (a + nc) < b.

We then have;

Lemma 0.12. Let ϵ be any positive infinitesimal, and let R_{f}(ϵ) be its value in R^{∗}, as guaranteed by Lemma 0.6. Then R_{f}(ϵ) is bounded, and, moreover, the standard part st(R_{f}(ϵ)) coincides with the usual definite integral;

∫_{a}^{b} f(x)dx

Proof. Let {S, T } be any real numbers such that;

S > max_{x∈[a,b]}f(x)

T < min_{x∈[a,b]}f(x)

Then a straightforward calculation shows that T (b − a) < R_{f}(c) < S(b − a), for 0 < c < (b − a). It follows easily, by formulating this statement in L_{R}, that T (b − a) < R_{f}(ϵ) < S(b − a), for any positive infinitesimal ϵ. Hence, the boundedness result follows. One can show easily that;

∫_{a}^{b} f(x)dx = lim_{n→∞}R_{f}(c_{n})

where c_{n} = \frac{(b−a)}{n}. It is a simple calculation, using the method of Lemma 0.3, to show that this must coincide with the standard part st(R_{f}(ϵ)), for any positive infinitesimal ϵ.

□

The following definition and lemma are then obvious;

Definition 0.13. Non-Standard Integration

Let f be a continuous function on the bounded interval [a, b], then we define its non-standard integral to be st(R_{f}(ϵ)), where ϵ is any positive infinitesimal.

Lemma 0.14. For a continuous function f on the bounded interval [a, b], the non-standard integral coincides with the usual (Riemann or Lebesgue)integral.

Proof. The proof follows immediately from the previous definition and Lemma 0.12. □

For future applications, it will be convenient to work with a particular class of non-standard extensions of R. Namely, we consider the structure R_{an} = (R, +, ., −, 0, 1, <; F) where F denotes the class of functions f̄: R^{m} → R of the following form;

For any analytic function f : U → R, where U is an open neighborhood of a product I_{1} × … × I_{m} of closed intervals, let;

f̄(x̄) = f(x̄) if x̄ ∈ I_{1} × … × I_{m} = 0 otherwise

The theory T h(R_{an}) is model complete, due to a theorem of Gabrielov, see [1]. We then make the following;

Definition 0.15. We define any structure (R^{∗}, +, ., −, 0, 1, <; F), that is an elementary extension of R_{an}, and contains an element ϵ such that;

0 < ϵ < r, for every r ∈ R

to be an analytic non-standard extension of R.

Remarks 0.16. Following the proof of Lemma 0.1, it is easily shown that analytic non-standard extensions exist. It is an easy exercise to show that we can obtain the remaining results of the paper, provided we restrict ourselves to analytic functions on a bounded open (or closed) interval in R.

References

[1] A. Gabrielov, Projections of Semi-Analytic Sets, Functional Anal. Appl. 2, (1968), 282-291.

[2] Abraham Robinson, Non-Standard Analysis, Princeton University Press, (1996)

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  E-mail address: tristam.depiro@unicam.it