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Uniqueness, Continuity, and Existence of Implicit Functions in Constructive Analysis

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Uniqueness, Continuity, and Existence of Implicit Functions in Constructive Analysis Uniqueness, Continuity, and Existence of Implicit Functions in Constructive Analysis Hannes Diener1 and Peter Schuster2 1 Universit¨atSiegen, Fachbereich 6: Mathematik Walter-Flex-Straße 3, 57072 Siegen, Germany [email protected] 2 Dipartimento di Filosofia, Universit`adegli Studi di Firenze Via Bolognese 52, 50139 Firenze, Italy [email protected] Abstract. We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit functions theorem. This leads not only to an a priori proof of continuity, but also to an alternative, fully constructive existence proof. 1 Introduction To show the differentiability of an implicit function one often relies on its continu- ity. The latter is mostly seen as a by-product of the not uncommon construction of the implicit function as the limit of a uniformly convergent sequence of con- tinuous functions. We now show that the continuity of the implicit function is prior to its existence, and thus independent of any particular construction. More specifically, we deduce the continuity from a quantitative strengthening of the uniqueness, which in turn follows from the hypotheses one needs to impose on the equation the implicit function is expected to satisfy. The same quantitative strengthening of uniqueness enables us to ultimately give an alternative existence proof for implicit functions that is fully constructive in Bishop's sense. We use ideas from [6], which loc.cit. have only been spelled out in the case of implicit functions with values in R. The existence proof given in [6] therefore can rely on reasoning by monotonicity, whereas in the general case|treated in this paper|of implicit functions with values in Rm we need to employ an extreme value argument. Similar considerations in related contexts can be found in Sections 3.3 and 3.4 of [10] during the course of the proof of the theorem on implicit functions via the inverse mapping theorem and Banach's fixed point theorem, respectively. We refer to [11, 20] for the implicit function theorem and the open mapping theorem in computable analysis `ala Weihrauch [19]. The predecessor [18] of the present paper essentially contains the same ma- terial as far as uniqueness and continuity of the implicit function are concerned. When it comes to proving existence, however, it follows the intrinsically classical argument that a continuous function on a compact set attains its minimum. This Andrej Bauer, Peter Hertling, Ker-I Ko (Eds.) 6th Int'l Conf. on Computability and Complexity in Analysis, 2009, pp. 131-140 http://drops.dagstuhl.de/opus/volltexte/2009/2265 132 Hannes Diener and Peter Schuster argument fails being practicable constructively, unless one adds the hypothesis that there quantitatively is at most one point at which the minimum can be attained. In fact, there is a heuristic principle valid [17] even in Bishop{style constructive mathematics without countable choice: if a continuous function on a complete metric space has approximate roots and in a quantitative manner at most one root, then it actually has a root. We may refer to [17] for more on this, including the principle's history with references. As a matter of fact, however, in the case of implicit functions the required ad- ditional hypothesis is contained in the quantitative variant of uniqueness which we find at our disposal anyway. Therefore, we only need to prove that for ev- ery parameter the given equation admits approximate solutions. Altogether we achieve the existence of an exact solution at every parameter and then, by the principle of unique choice, the existence of an implicit function: as the one and only function which assigns to every parameter the solution uniquely determined by this parameter. The present paper as a whole is conceived in the realm of Bishop's construc- tive mathematics [4, 5, 7, 8]. Compared with the|so-called classical|customary way of doing mathematics, the principal characteristic of the framework created by Bishop is the exclusive use of intuitionistic logic, which allows one to view Bishop's setting as a generalisation of classical mathematics [13]. Moreover, we follow [14] in doing constructive mathematics `ala Bishop without countable choice, also inasmuch as we understand real numbers as located Dedekind cuts. In particular, the so-called cotransitivity property \if x < y, then x < z or z < y" amounts to say that the Dedekind cut z is located whenever x; y are rational numbers, and follows by approximation in the general case. Avoiding countable choice is further indispensable, because we want our work to be expressible in constructive Zermelo{Fraenkel set theory (CZF) as begun in [1]: countable choice does not belong to CZF. Details on this and on CZF in general can be found in [2, 12]. We will, however, use the principle of unique choice, sometimes called the principle of non{choice. By the functions{as{graphs paradigm common to set theory, unique choice is trivially in CZF. 2 Preliminaries We first recall that in Bishop's setting every differentiable function comes with a continuous derivative [5, Chapter 2, Section 5]. In other words, for Bishop every differentiable function is by definition continuously differentiable. We nonethe- less keep speaking of continously differentiable functions, also to facilitate any reading by a classically trained person. Note in this context that in Bishop's framework continuity means uniform continuity on every compact (that is, to- tally bounded and complete) subset of the domain; see [15] for a discussion of this. Secondly, although in the work of Bishop and of his followers there barely is any talk of (partial or total) differentiability for functions of several real vari- ables, we do not develop this concept in the present paper either. According to Implicit Functions in Constructive Analysis 133 our opinion it is in order to take this for granted: the task of checking the classical route as far as necessary can indeed be performed in a relatively straightforward way, and is sometimes simplified by Bishop's assumption of the automatic con- tinuity of the derivative. For the lack of appropriate references in the constructive literature we next transfer two facts from real analysis. With Theorem 5.4 and Theorem 6.8 of [5, Chapter 2] at hand the standard proofs indeed go through constructively. (For instance, the proofs of Satz 5 and of its Corollar given in [9, I, x6] require only one addendum to the proof of the Hilfssatz: for all K; L 2 R with L > 0 the 2 implication \if K 6 KL, then K 6 L" is also constructively valid. To verify 2 this, assume that K 6 KL; it suffices to prove that K < L + " for every " > 0. 2 For each " > 0 either 0 < K or K < ". In the former case, multiplying K 6 KL by 1=K > 0 yields K 6 L; in the latter case we have K < L + " because L > 0.) Lemma 1. Let g : W ! Rn be a continuously differentiable mapping on an open set W ⊆ Rm, and c; d 2 W . If the line segment between c and d lies entirely in W , then Z 1 g(d) − g(c) = Dg(c + t(d − c))dt · (d − c) : 0 Corollary 2. Under the hypotheses of Lemma 1 we have kg(d) − g(c)k 6 sup kDg(c + t(d − c))dtk · kd − ck : t2[0;1] While Lemma 5.5 of [5, Chapter 2] is an approximative alternative of Rolle's theorem, our next lemma is a strong variant of the contrapositive. Lemma 3. Let h :[c; d] ! R be continuously differentiable, and assume that there is r > 0 such that h0(x) > r for all x 2 [c; d]. Furthermore assume that c < d. Then h(c) < h(d). r(d−c) Proof. Assume that h(d) − h(c) < 4 . By the mean value theorem [5, Theo- rem 5.6] there exists ξ 2 [c; d] such that r(d − c) jh0(ξ)(d − c) − (h(d) − h(c))j < : 2 Then h0(ξ)(d − c) − (h(d) − h(c)) > r(d − c) − (h(d) − h(c)) r(d − c) > r(d − c) − 4 r(d − c) > : 2 Hence we get a contradiction, and thus h(d) − h(c) > 0. ut 134 Hannes Diener and Peter Schuster The last lemma in this section is an approximative substitute for the classical result that if a differentiable function attains its minimum at a point in the interior of a compact set, then the gradient of that function vanishes at this point. Lemma 4. Let W ⊆ Rn be an open neighbourhood of [0; 1]n and h : W ! R a continuously differentiable function. If there is a point ξ 2 [0; 1]n and s > 0 such that h(x) > h(ξ) + s (1) for all x 2 @[0; 1]n, then for every " > 0 there exists y 2 [0; 1]n such that krh(y)k < ". Proof. For convenience we will use the supremum norm on Rn throughout this proof. Choose N 2 N such that for x; y 2 [0; 1]n, if ky − xk < 2−N then both " krh(x) − rh(y)k < (2) 4 and s jh(x) − h(y)j < : (3) 2 Let i i G = 1 ;:::; n :(i ; : : : ; i ) 2 n \ [0; 1]n: 2N 2N 1 n N ± −N For any x 2 G and i 6 n, let xi denote the point x±2 ei|i.e. the neighbouring point of x in G in the positive/negative direction of the ith coordinate. For any + − x 2 G and i 6 n fix λx;i 2 {−1; 0; 1g and λx;i 2 {−1; 0; 1g, such that + @h −(N+1) 3" λx;i = 0 ) (x + 2 ei) < ; @xi 4 + @h −(N+1) " λx;i = −1 ) (x + 2 ei) < − ; @xi 2 + @h −(N+1) " λx;i = 1 ) (x + 2 ei) > ; @xi 2 − @h −(N+1) 3" λx;i = 0 ) (x − 2 ei) < ; @xi 4 − @h −(N+1) " λx;i = −1 ) (x − 2 ei) > ; @xi 2 − @h −(N+1) " λx;i = 1 ) (x − 2 ei) < − : @xi 2 + + Notice that if λx;i = −1 then for all y 2 [x; xi ] @h " (y) < − ; @xi 4 Implicit Functions in Constructive Analysis 135 and therefore, by Lemma 3, + h(x) > h(xi ): (4) − Similarly, when λx;i = −1, we obtain @h " (y) > @xi 4 − for all y 2 [xi ; x], and then − h(x) > h(xi ): (5) Furthermore notice that, by continuity and (2), + − @h if λx;i 2 f0; 1g and λx;i 2 f0; 1g, then (x) < ": (6) @xi Next, because of (3), we can find x0 2 G such that jh(x0) − h(ξ)j < s.
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