Abstract

We study the properties of locally uniformly differentiable functions on ๐’ฉ, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are ๐ถ1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

1. Introduction

We start this section by reviewing some basic terminology and facts about non-Archimedean fields. So let ๐น be an ordered non-Archimedean field extension of โ„. We introduce the following terminology.

Definition 1.1 (~, โ‰ˆ, โ‰ช, ๐‘†๐น, ๐œ†). For ๐‘ฅ,๐‘ฆโˆˆ๐นโˆ—โˆถ=๐นโงต{0}, we say ๐‘ฅโˆผ๐‘ฆ if there exist ๐‘›,๐‘šโˆˆโ„• such that ๐‘›|๐‘ฅ|>|๐‘ฆ| and ๐‘š|๐‘ฆ|>|๐‘ฅ|, where |โ‹…| denotes the usual absolute value on ๐น: ๎‚ป|๐‘ฅ|=๐‘ฅif๐‘ฅโ‰ฅ0,โˆ’๐‘ฅif๐‘ฅ<0.(1.1) For nonnegative ๐‘ฅ,๐‘ฆโˆˆ๐น, one says that ๐‘ฅ is infinitely smaller than ๐‘ฆ and write ๐‘ฅโ‰ช๐‘ฆ if ๐‘›๐‘ฅ<๐‘ฆ for all ๐‘›โˆˆโ„•, and we say that ๐‘ฅ is infinitely small if ๐‘ฅโ‰ช1 and ๐‘ฅ is finite if ๐‘ฅโˆผ1; finally, we say that ๐‘ฅ is approximately equal to ๐‘ฆ and write ๐‘ฅโ‰ˆ๐‘ฆ if ๐‘ฅโˆผ๐‘ฆ and |๐‘ฅโˆ’๐‘ฆ|โ‰ช|๐‘ฅ|. We also set ๐œ†(๐‘ฅ)=[๐‘ฅ], the class of ๐‘ฅ under the equivalence relation ~.

The set of equivalence classes ๐‘†๐น (under the relation ~) is naturally endowed with an addition via [๐‘ฅ]+[๐‘ฆ]=[๐‘ฅโ‹…๐‘ฆ] and an order via [๐‘ฅ]<[๐‘ฆ] if |๐‘ฆ|โ‰ช|๐‘ฅ| (or |๐‘ฅ|โ‰ซ|๐‘ฆ|), both of which are readily checked to be well defined. It follows that (๐‘†๐น,+,<) is an ordered group, often referred to as the Hahn group or skeleton group, whose neutral element is [1], the class of 1. It follows from the previous part that the projection ๐œ† from ๐นโˆ— to ๐‘†๐น is a valuation.

The theorem of Hahn [2] provides a complete classification of non-Archimedean extensions of โ„ in terms of their skeleton groups. In fact, invoking the axiom of choice, it is shown that the elements of any such field ๐น can be written as formal power series over its skeleton group ๐‘†๐น with real coefficients, and the set of appearing exponents forms a well-ordered subset of ๐‘†๐น.

From general properties of formal power series fields [3, 4], it follows that if ๐‘†๐น is divisible, then ๐น is real closed; that is, every positive element of ๐น is a square in ๐น and every polynomial of odd degree over ๐น has at least one root in ๐น. For a general overview of the algebraic properties of formal power series fields, we refer to the comprehensive overview by Ribenboim [5], and for an overview of the related valuation theory, the book by Krull [6]. A thorough and complete treatment of ordered structures can also be found in [7].

Throughout this paper, we will denote by ๐’ฉ any totally ordered non-Archimedean field extension of โ„ that is complete in the order topology and whose skeleton group ๐‘†๐’ฉ is Archimedean, that is, a subgroup of โ„. The coefficient of the ๐‘žth power in the Hahn representation of a given ๐‘ฅ will be denoted by ๐‘ฅ[๐‘ž], and the number ๐‘‘ will be defined by ๐‘‘[1]=1 and ๐‘‘[๐‘ž]=0 for ๐‘žโ‰ 1. It is easy to check that, for ๐‘žโˆˆ๐‘†๐’ฉ, 0<๐‘‘๐‘žโ‰ช1 if and only if ๐‘ž>0, and ๐‘‘๐‘žโ‰ซ1 if and only if ๐‘ž<0; moreover, ๐‘ฅโ‰ˆ๐‘ฅ[๐œ†(๐‘ฅ)]๐‘‘๐œ†(๐‘ฅ) for all ๐‘ฅโ‰ 0.

The smallest such field ๐’ฉ is the field ๐ฟ of the formal Laurent series whose skeleton group is ๐‘†๐ฟ=โ„ค, and the smallest such field that is also real closed is the Levi-Civita field โ„›, first introduced in [8, 9]. In this latter case ๐‘†โ„›=โ„š, and for any element ๐‘ฅโˆˆโ„›, the set of exponents in the Hahn representation of ๐‘ฅ is a left-finite subset of โ„š; that is, below any rational bound ๐‘Ÿ there are only finitely many exponents. The Levi-Civita field โ„› is of particular interest because of its practical usefulness. Since the supports of the elements of โ„› (when viewed as maps from ๐‘†โ„›=โ„š into โ„) are left-finite, it is possible to represent these numbers on a computer [1]. Having infinitely small numbers, the errors in classical numerical methods can be made infinitely small and hence irrelevant in all practical applications. One such application is the computation of derivatives of real functions representable on a computer [10], where both the accuracy of formula manipulators and the speed of classical numerical methods are achieved. For a review of the Levi-Civita field โ„›, see [11, 12] and the references therein.

In the wider context of valuation theory, it is interesting to note that the topology induced by the order on ๐’ฉ is the same as that introduced via the valuation ๐œ†, as shown in Remark 1.2. It follows therefore that the field ๐’ฉ is just a special case of the class of fields discussed in [13].

Remark 1.2. The mapping ฮ›โˆถ๐’ฉร—๐’ฉโ†’โ„, given by ฮ›(๐‘ฅ,๐‘ฆ)=exp(โˆ’๐œ†(๐‘ฅโˆ’๐‘ฆ)), is an ultrametric distance (and hence a metric); the valuation topology it induces is equivalent to the order topology (we will use ๐œ๐‘ฃ to denote either one of the two topologies in the rest of the paper). If ๐ด is an open set in the order topology and ๐‘Žโˆˆ๐ด, then there exists ๐‘Ÿ>0 in ๐’ฉ such that, for all ๐‘ฅโˆˆ๐’ฉ, |๐‘ฅโˆ’๐‘Ž|<๐‘Ÿโ‡’๐‘ฅโˆˆ๐ด. Let ๐‘™=exp(โˆ’๐œ†(๐‘Ÿ)); then we also have that, for all ๐‘ฅโˆˆ๐’ฉ, ฮ›(๐‘ฅ,๐‘Ž)<๐‘™โ‡’๐‘ฅโˆˆ๐ด, and hence ๐ด is open with respect to the valuation topology. The other direction of the equivalence of the topologies follows analogously.

In this paper, we will study the properties of locally uniformly differentiable functions on ๐’ฉ, thus expanding on the work done in [14]. In particular, we will show that this class of functions is closed under addition, multiplication, and composition of functions. Then we will state and prove a more general version of the inverse function theorem than that proved in [14], as well as a local intermediate value theorem for ๐’ฉ-valued locally uniformly differentiable functions on ๐’ฉ. The stronger condition (local uniform differentiability) on the function than that of the real case is needed for the proof of both theorems because of the total disconnectedness of the field ๐’ฉ in the order topology.

2. Preliminaries

In this section we review some of the topological properties of the field ๐’ฉ which helps the reader understand the differences between ๐’ฉ and โ„. We begin this with the following definition.

Definition 2.1. Let ๐ดโŠ‚๐’ฉ. Then we say that ๐ด is compact in (๐’ฉ,๐œ๐‘ฃ) if every open cover of ๐ด in (๐’ฉ,๐œ๐‘ฃ) has a finite subcover.

Remark 2.2. Since ๐œ๐‘ฃ is induced by a metric on ๐’ฉ, namely, the valuation metric ฮ› mentioned in the Introduction, it follows by the Borel-Lebesgue Theorem (see, e.g., [15, Section 9.2]) that ๐ด is compact in (๐’ฉ,๐œ๐‘ฃ) if and only if ๐ด is sequentially compact.

Theorem 2.3. (๐’ฉ,๐œ๐‘ฃ) is a totally disconnected topological space. It is Hausdorff and nowhere locally compact. There are no countable bases. The topology induced to โ„ is the discrete topology.

Proof. Let ๐ดโŠ‚๐’ฉ contain more than one point, and let ๐‘Žโ‰ ๐‘ in ๐ด be given. Without loss of generality, we may assume that ๐‘Ž<๐‘. Let ๐บ1={๐‘ฅโˆˆ๐’ฉโˆถ|๐‘ฅโˆ’๐‘Ž|โ‰ช๐‘โˆ’๐‘Ž},๐บ2=๐’ฉโงต๐บ1.(2.1) Then ๐บ1 and ๐บ2 are disjoint and open in (๐’ฉ,๐œ๐‘ฃ), ๐‘Žโˆˆ๐บ1โˆฉ๐ด and ๐‘โˆˆ๐บ2โˆฉ๐ด, and ๐ดโŠ‚๐บ1โˆช๐บ2=๐’ฉ. This shows that any subset of (๐’ฉ,๐œ๐‘ฃ) containing more than one point is disconnected, and hence (๐’ฉ,๐œ๐‘ฃ) is totally disconnected. It follows that (๐’ฉ,๐œ๐‘ฃ) is Hausdorff. That (๐’ฉ,๐œ๐‘ฃ) is Hausdorff also follows from the fact that it is a metric space ([16, p. 66, Problem 7(a)]).
To prove that (๐’ฉ,๐œ๐‘ฃ) is nowhere locally compact, let ๐‘ฅโˆˆ๐’ฉ be given and let ๐‘ˆ be a neighborhood of ๐‘ฅ. We show that the closure ๐‘ˆ of ๐‘ˆ is not compact. Let ๐œ–>0 in ๐’ฉ be such that (๐‘ฅโˆ’๐œ–,๐‘ฅ+๐œ–)โŠ‚๐‘ˆ and consider the sets ๐‘€โˆ’1๐‘€={๐‘ฆโˆˆ๐’ฉโˆถ๐‘ฆ<๐‘ฅor๐‘ฆโˆ’๐‘ฅโ‰ซ๐‘‘โ‹…๐œ–},๐‘›=(๐‘ฅ+(๐‘›โˆ’1)๐‘‘โ‹…๐œ–,๐‘ฅ+(๐‘›+1)๐‘‘โ‹…๐œ–)for๐‘›=0,1,2,โ€ฆ,(2.2) where ๐‘‘ is the infinitely small positive number defined in the introduction. Then it is easy to check that ๐‘€๐‘› is open in (๐’ฉ,๐œ๐‘ฃ) for all ๐‘›โ‰ฅโˆ’1, and โ‹ƒโˆž๐‘›=โˆ’1๐‘€๐‘›=๐’ฉ; in particular, โ‹ƒ๐‘ˆโŠ‚โˆž๐‘›=โˆ’1๐‘€๐‘›. But it is impossible to select finitely many of the ๐‘€๐‘›โ€™s to cover ๐‘ˆ because each of the infinitely many elements ๐‘ฅ+๐‘›๐‘‘โ‹…๐œ– of ๐‘ˆ, ๐‘›=โˆ’1,0,1,2,โ€ฆ, is contained only in the set ๐‘€๐‘›.
There cannot be any countable bases because the uncountably many open sets ๐‘€๐‘‹=(๐‘‹โˆ’๐‘‘,๐‘‹+๐‘‘), with ๐‘‹โˆˆโ„, are disjoint. The open sets induced on โ„ by the sets ๐‘€๐‘‹ are just the singletons {๐‘‹}. Thus, in the induced topology, all sets are open and the induced topology is therefore discrete.

As an immediate consequence of the fact that (๐’ฉ,๐œ๐‘ฃ) is totally disconnected, it follows that, for any ๐‘ฅ0โˆˆ๐’ฉ, the connected component of ๐‘ฅ0 is {๐‘ฅ0}. Moreover, there are sets that are both open and closed, as we will show hereinafter.

Definition 2.4. Let ฮฉโŠ‚๐’ฉ. Then we say that ฮฉ is clopen in (๐’ฉ,๐œ๐‘ฃ) if it is both open and closed.

Proposition 2.5. For any ๐‘ฅ0โˆˆ๐’ฉ and for any ๐‘Ž>0 in ๐’ฉ, the set ฮฉ={๐‘ฅโˆถ|๐‘ฅโˆ’๐‘ฅ0|โ‰ช๐‘Ž} is clopen in (๐’ฉ,๐œ๐‘ฃ).

Proof. Let ๐‘ฅโˆˆฮฉ be given. For all ๐‘ฆโˆˆ(๐‘ฅโˆ’๐‘Ž๐‘‘,๐‘ฅ+๐‘Ž๐‘‘), we have that ||๐‘ฆโˆ’๐‘ฅ0||โ‰ค||||+||๐‘ฆโˆ’๐‘ฅ๐‘ฅโˆ’๐‘ฅ0||||<๐‘Ž๐‘‘+๐‘ฅโˆ’๐‘ฅ0||โ‰ช๐‘Ž.(2.3) Thus, (๐‘ฅโˆ’๐‘Ž๐‘‘,๐‘ฅ+๐‘Ž๐‘‘)โŠ‚ฮฉ and hence ฮฉ is open.
Now let ๐‘ฅโˆˆ๐’ฉโงตฮฉ. Then for all ๐‘ฆโˆˆ(๐‘ฅโˆ’๐‘Ž๐‘‘,๐‘ฅ+๐‘Ž๐‘‘), we have that ||๐‘ฆโˆ’๐‘ฅ0||โ‰ฅ||||||โˆ’||๐‘ฆโˆ’๐‘ฅ๐‘ฅโˆ’๐‘ฅ0||||โ‰ˆ||๐‘ฅโˆ’๐‘ฅ0||ยจยจโ‰ช๐‘Ž;(2.4)
so ๐‘ฆโˆˆ๐’ฉโงตฮฉ. Thus, (๐‘ฅโˆ’๐‘Ž๐‘‘,๐‘ฅ+๐‘Ž๐‘‘)โŠ‚๐’ฉโงตฮฉ and hence ๐’ฉโงตฮฉ is open. That is, ฮฉ is closed.

Similarly we can show that the sets {๐‘ฅโˆถ|๐‘ฅโˆ’๐‘ฅ0|โˆผ๐‘Ž} and {๐‘ฅโˆถ|๐‘ฅโˆ’๐‘ฅ0|โ‰ˆ๐‘Ž} are clopen for any ๐‘ฅ0โˆˆ๐’ฉ and any ๐‘Ž>0 in ๐’ฉ.

Proposition 2.6. Let ๐‘ฅ0โˆˆ๐’ฉ be given and let ฮฉ be a neighborhood of ๐‘ฅ0. Then there is a clopen set ๐ฟ such that ๐‘ฅ0โˆˆ๐ฟโŠ‚ฮฉ.

Proof. Let ๐œ–>0 in ๐’ฉ be such that (๐‘ฅ0โˆ’๐œ–,๐‘ฅ0+๐œ–)โŠ‚ฮฉ. Let ๐ฟ={๐‘ฅโˆˆ๐’ฉโˆถ|๐‘ฅโˆ’๐‘ฅ0|โ‰ช๐œ–}. Then ๐ฟ is clopen by Proposition 2.5 and ๐‘ฅ0โˆˆ๐ฟโŠ‚(๐‘ฅ0โˆ’๐œ–,๐‘ฅ0+๐œ–)โŠ‚ฮฉ.

It follows that the clopen sets form a base for the order topology. Moreover, the quasi-component of any ๐‘ฅ0โˆˆ๐’ฉ is {๐‘ฅ0}.

As an immediate consequence of the fact that (๐’ฉ,๐œ๐‘ฃ) is nowhere locally compact, we obtain the following result.

Corollary 2.7. For all ๐‘Ž<๐‘ in ๐’ฉ, none of the intervals (๐‘Ž,๐‘), (๐‘Ž,๐‘], [๐‘Ž,๐‘), or [๐‘Ž,๐‘] are compact in (๐’ฉ,๐œ๐‘ฃ).

Since ๐œ๐‘ฃ is induced on ๐’ฉ by the order, we define boundedness of a set in (๐’ฉ,๐œ๐‘ฃ) as follows.

Definition 2.8. Let ๐ดโŠ‚๐’ฉ. Then we say that ๐ด is bounded in (๐’ฉ,๐œ๐‘ฃ) if there exists ๐‘€>0 in ๐’ฉ such that |๐‘ฅ|โ‰ค๐‘€ for all ๐‘ฅโˆˆ๐ด.

Proposition 2.9. Let ๐ด be compact in (๐’ฉ,๐œ๐‘ฃ). Then ๐ด is closed and bounded in (๐’ฉ,๐œ๐‘ฃ). Moreover, ๐ด has an empty interior in (๐’ฉ,๐œ๐‘ฃ); that is, int(๐ด)โˆถ={๐‘Žโˆˆ๐ดโˆถโˆƒ๐‘Ÿ>0in๐’ฉโˆ‹(๐‘Žโˆ’๐‘Ÿ,๐‘Ž+๐‘Ÿ)โŠ‚๐ด}=โˆ….(2.5)

Proof. That ๐ด is closed in (๐’ฉ,๐œ๐‘ฃ) follows from the fact that (๐’ฉ,๐œ๐‘ฃ) is a Hausdorff topological space and ๐ด is compact in (๐’ฉ,๐œ๐‘ฃ) (see [17, p. 36]).
Now we show that ๐ด is bounded in (๐’ฉ,๐œ๐‘ฃ). For each ๐‘›โˆˆโ„•, let ๐บ๐‘›=(โˆ’๐‘‘โˆ’๐‘›,๐‘‘โˆ’๐‘›). Then, for each ๐‘›โˆˆโ„•, ๐บ๐‘› is open in (๐’ฉ,๐œ๐‘ฃ). Moreover, since the skeleton group of ๐’ฉ is Archimedean it follows that โ‹ƒ๐ดโŠ‚๐‘›โˆˆโ„•๐บ๐‘›=๐’ฉ. Since ๐ด is compact in (๐’ฉ,๐œ๐‘ฃ), we can choose a finite subcover; thus, there is ๐‘šโˆˆโ„• and there exist ๐‘—1<๐‘—2<โ‹ฏ<๐‘—๐‘š in โ„• such that๐ดโŠ‚๐‘šโ‹ƒ๐‘™=1๐บ๐‘—๐‘™=๐บ๐‘—๐‘š=๎€ทโˆ’๐‘‘โˆ’๐‘—๐‘š,๐‘‘โˆ’๐‘—๐‘š๎€ธ.(2.6) It follows that |๐‘ฅ|<๐‘‘โˆ’๐‘—๐‘š for all ๐‘ฅโˆˆ๐ด, and hence ๐ด is bounded in (๐’ฉ,๐œ๐‘ฃ).
Finally, we show that int(๐ด)=โˆ…. Assume not. Then there exist ๐‘Ž<๐‘ in ๐ด such that [๐‘Ž,๐‘]โŠ‚๐ด. Since [๐‘Ž,๐‘] is a closed subset of the compact set ๐ด, it follows that [๐‘Ž,๐‘] is compact in (๐’ฉ,๐œ๐‘ฃ), which contradicts Corollary 2.7.

The following examples show that there are (countably infinite) closed and bounded sets that are not compact while there are uncountable sets that are compact in (๐’ฉ,๐œ๐‘ฃ).

Example 2.10. Let ๐ด=[0,1]โˆฉโ„š. Then clearly, ๐ด is countably infinite and bounded in (๐’ฉ,๐œ๐‘ฃ). We show that ๐ด is closed in (๐’ฉ,๐œ๐‘ฃ). Let ๐‘ฅโˆˆ๐’ฉโงต๐ด be given and let ๐บ0=(๐‘ฅโˆ’๐‘‘,๐‘ฅ+๐‘‘). If ๐บ0โˆฉ๐ดโ‰ โˆ…, then there exists ๐‘žโˆˆ๐ด such that ๐บ0โˆฉ๐ด={๐‘ž}. Let ๐‘Ÿ=|๐‘žโˆ’๐‘ฅ| and let ๐บ=(๐‘ฅโˆ’๐‘Ÿ,๐‘ฅ+๐‘Ÿ). Then ๐บ is open in (๐’ฉ,๐œ๐‘ฃ) and ๐บโˆฉ๐ด=โˆ…. Thus, ๐’ฉโงต๐ด is open, and hence ๐ด is closed in (๐’ฉ,๐œ๐‘ฃ).
Next we show that ๐ด is not compact in (๐’ฉ,๐œ๐‘ฃ). For each ๐‘žโˆˆ๐ด, let ๐บ๐‘ž=(๐‘žโˆ’๐‘‘,๐‘ž+๐‘‘). Then ๐บ๐‘ž is open in (๐’ฉ,๐œ๐‘ฃ) for each ๐‘ž and โ‹ƒ๐ดโŠ‚๐‘žโˆˆ๐ด๐บ๐‘ž, but we cannot select a finite subcover since each ๐‘กโˆˆ๐ด is contained only in ๐บ๐‘ก.

Example 2.11. Let ๐ถ๐’ฉ denote the Cantor-like set constructed in the same way as the standard real Cantor set ๐ถ; but instead of deleting the middle third, we delete from the middle an open interval (1โˆ’2๐‘‘) times the size of each of the closed subintervals of [0,1] at each step of the construction. Then ๐ถ๐’ฉ is compact in (๐’ฉ,๐œ๐‘ฃ).
It turns out that if we view ๐’ฉ as an infinite dimensional vector space over โ„, then ๐œ๐‘ฃ is not a vector topology; that is, (๐’ฉ,๐œ๐‘ฃ) is not a linear topological space.

Theorem 2.12. ๐œ๐‘ฃ is not a vector topology.

Proof. Assume to the contrary that (๐’ฉ,๐œ๐‘ฃ) is a vector topology. Then, by continuity of scalar multiplication, there exists an open set ๐‘‚โ„โŠ‚โ„ and there exists an open set ๐‘‚๐’ฉโŠ‚๐’ฉ such that ๐›ผ๐‘ฅโˆˆ(1โˆ’๐‘‘,1+๐‘‘) for all ๐›ผโˆˆ๐‘‚โ„ and for all ๐‘ฅโˆˆ๐‘‚๐’ฉ. Let ๐›ผ0โˆˆ๐‘‚โ„ and ๐‘ฅ0โˆˆ๐‘‚๐’ฉ be given. Since ๐‘‚โ„ is open, there exists ๐‘Ÿ>0 in โ„ such that (๐›ผ0โˆ’2๐‘Ÿ,๐›ผ0+2๐‘Ÿ)โŠ‚๐‘‚โ„. Hence ๐›ผ0๐‘ฅ0๎€ท๐›ผโˆˆ(1โˆ’๐‘‘,1+๐‘‘),0๎€ธ๐‘ฅ+๐‘Ÿ0โˆˆ(1โˆ’๐‘‘,1+๐‘‘).(2.7) Thus, ๐‘Ÿ||๐‘ฅ0||=||๎€ท๐›ผ0๎€ธ๐‘ฅ+๐‘Ÿ0โˆ’๐›ผ0๐‘ฅ0||<2๐‘‘,(2.8) which contradicts the fact that ๐‘Ÿ|๐‘ฅ0|โ‰ซ2๐‘‘, since both ๐‘Ÿ and |๐‘ฅ0| are finite and ๐‘‘ is infinitely small.

Since any normed vector space, with the metric topology induced by its norm, is a linear topological space ([18, Proposition III.1.3]), we readily infer from Theorem 2.12 that there can be no norm on ๐’ฉ that would induce the same topology as ๐œ๐‘ฃ on ๐’ฉ.

We finish this section with the following criterion for convergence for an infinite series, which does not hold for real numbers series.

Proposition 2.13. For each ๐‘›โˆˆโ„•, let ๐‘ฅ๐‘› be an element of ๐’ฉ. Then the series โˆ‘โˆž๐‘›=1๐‘ฅ๐‘› converges if and only if the sequence (๐‘ฅ๐‘›) converges to zero.

Proof. Assume that โˆ‘โˆž๐‘›=1๐‘ฅ๐‘› converges, and let (๐‘ฆ๐‘›) denote the sequence of partial sums of the series: ๐‘ฆ๐‘›=โˆ‘๐‘›๐‘–=1๐‘ฅ๐‘–. Thus (๐‘ฆ๐‘›) converges and hence it is Cauchy. Now let ๐œ–>0 in ๐’ฉ be given. Then there exists ๐‘โˆˆโ„• such that for each ๐‘›,๐‘š>๐‘ we have that |๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘š|<๐œ–. It follows that |๐‘ฅ๐‘›+1|=|๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›|<๐œ– for all ๐‘›>๐‘ or, equivalently, |๐‘ฅ๐‘›|<๐œ– for all ๐‘›>๐‘+1. Hence the sequence (๐‘ฅ๐‘›) converges to zero.
Now assume that the sequence (๐‘ฅ๐‘›) converges to zero. Let (๐‘ฆ๐‘›) be the sequence of partial sums of the series โˆ‘โˆž๐‘›=1๐‘ฅ๐‘› and let ๐œ–>0 be given in ๐’ฉ. Then there is an ๐‘โˆˆโ„• such that |๐‘ฅ๐‘›|<๐‘‘๐œ– for all ๐‘›>๐‘. Thus, for all ๐‘›>๐‘š>๐‘, we have that |๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘šโˆ‘|=|๐‘›๐‘–=๐‘š+1๐‘ฅ๐‘–|<(๐‘šโˆ’๐‘›)๐‘‘๐œ–โ‰ช๐œ–. Thus (๐‘ฆ๐‘›) is Cauchy, and hence โˆ‘โˆž๐‘›=1๐‘ฅ๐‘› converges since (๐’ฉ,๐œ๐‘ฃ) is complete.

3. Local Uniform Continuity

In this section we introduce the concept of local uniform continuity of a function from a subset of ๐’ฉ to ๐’ฉ and study properties of such functions that will be relevant to our discussion of locally uniformly differentiable functions later in Section 4. We start with the following definitions.

Definition 3.1. Let ๐ดโŠ‚๐’ฉ, let ๐‘“โˆถ๐ดโ†’๐’ฉ, and let ๐‘ฅ0โˆˆ๐ด be given. Then one says that ๐‘“ is continuous at ๐‘ฅ0 if for every ๐œ–>0 there exists ๐›ฟ>0 such that |๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅ0)|<๐œ– whenever ๐‘ฅโˆˆ๐ด and |๐‘ฅโˆ’๐‘ฅ0|<๐›ฟ.

Definition 3.2. Let ๐ดโŠ‚๐’ฉ, let ๐‘“โˆถ๐ดโ†’๐’ฉ, and let ๐‘ฅ0โˆˆ๐ด be given. Then one says that ๐‘“ locally uniformly continuous at ๐‘ฅ0 if there is a neighborhood ฮฉ of ๐‘ฅ0 in ๐ด such that ๐‘“ is uniformly continuous on ฮฉ. That is, for every ๐œ–>0, there exists ๐›ฟ>0 such that |๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)|<๐œ– whenever ๐‘ฅ,๐‘ฆโˆˆฮฉ and |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ.

Exactly as in real calculus, one can easily show that if ๐‘“,๐‘”โˆถ๐ดโ†’๐’ฉ are (locally uniformly) continuous at ๐‘ฅ0โˆˆ๐ด and if ๐›ผโˆˆ๐’ฉ, then ๐‘“+๐›ผ๐‘” is (locally uniformly) continuous at ๐‘ฅ0. Moreover, if ๐ด,๐ตโŠ‚๐’ฉ and if ๐‘“โˆถ๐ดโ†’๐ต is (locally uniformly) continuous at ๐‘ฅ0โˆˆ๐ด and ๐‘”โˆถ๐ตโ†’๐’ฉ is (locally uniformly) continuous at ๐‘“(๐‘ฅ0), then ๐‘”โˆ˜๐‘“โˆถ๐ดโ†’๐’ฉ is (locally uniformly) continuous at ๐‘ฅ0.

Lemma 3.3. Let ๐‘ฅ0โˆˆ๐’ฉ be given and let ฮฉ be a neighborhood of ๐‘ฅ0. Then there exist sequences (๐‘ฅ๐‘›) and (๐‘ฆ๐‘›) as well as mutually disjoint clopen sets ๐‘ˆ๐‘› and ๐‘ˆ0 and a continuous function ๐‘“ such that(1)the set {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•} has no limit point;(2)lim๐‘›โ†’โˆž|๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›|=0;(3)๐‘ฅ0โˆ‰{๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•};(4)๐‘ฅ๐‘›โˆˆ๐‘ˆ๐‘› for each ๐‘›โˆˆโ„•โˆช{0};(5)๐‘ฆ๐‘›โˆ‰๐‘ˆ๐‘š for any ๐‘›,๐‘šโˆˆโ„•;(6)๐‘“(โˆช๐‘›โˆˆโ„•๐‘ˆ๐‘›)={1}, ๐‘“(๐’ฉโงตโˆช๐‘›โˆˆโ„•๐‘ˆ๐‘›)={0}.

Proof. Since ๐’ฉ is not compact, there is a sequence (๐‘ฅ๐‘›)โŠ‚ฮฉ that has no limit point in ๐’ฉ. Without loss of generality, we can take ๐‘ฅ0โˆ‰{๐‘ฅ๐‘›โˆถ๐‘›โˆˆโ„•} since {๐‘ฅ๐‘›โˆถ๐‘›โˆˆโ„•}โงต{๐‘ฅ0} will still have no limit point. For each ๐‘›โˆˆโ„•, let ๐‘ฆ๐‘›โˆˆ๐’ฉ be such that |๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›|<๐‘‘๐‘› and ๐‘ฆ๐‘›โ‰ ๐‘ฅ๐‘š for any ๐‘›โˆˆโ„• and ๐‘šโˆˆโ„•โˆช{0}. Since (๐‘‘๐‘›) is a null sequence in ๐’ฉ, it follows that |๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›|โ†’0 as ๐‘›โ†’โˆž. Assume that {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•} has a limit point in ๐’ฉ and let ๐‘ be such a limit point. Since (๐‘ฅ๐‘›) has no limit point, there is an ๐œ–>0 such that (๐‘โˆ’๐œ–,๐‘+๐œ–)โˆฉ{๐‘ฅ๐‘›โˆถ๐‘›โˆˆโ„•}=โˆ…. There exists ๐‘โˆˆโ„• such that, for all ๐‘›โ‰ฅ๐‘, |๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›|<๐œ–/2. Since ๐‘ is the limit point of {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•}, there must be ๐‘€>๐‘ such that ๐‘ฆ๐‘€โˆˆ(๐‘โˆ’๐œ–/2,๐‘+๐œ–/2). But then |๐‘ฅ๐‘€โˆ’๐‘|โ‰ค|๐‘ฅ๐‘€โˆ’๐‘ฆ๐‘€|+|๐‘ฆ๐‘€โˆ’๐‘|<๐œ–/2+๐œ–/2=๐œ–. This is a contradiction. Hence {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•} has no limit point.
Since {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•} has no limit point, there exist ๐‘ˆโ€ฒ๐‘› and ๐‘ˆ๎…ž0 such that ๐‘ฅ0โˆˆ๐‘ˆ๎…ž0, {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•}โˆฉ๐‘ˆ๎…ž๐‘›={๐‘ฅ๐‘›} and {๐‘ฅ๐‘›,๐‘ฆ๐‘›โˆถ๐‘›โˆˆโ„•}โˆฉ๐‘ˆ๎…ž0=โˆ…. For each ๐‘›โˆˆโ„•โˆช{0} let ๐œ–๐‘›>0 in ๐’ฉ be such that ๐œ–๐‘›โ‰ค๐‘‘๐‘› and (๐‘ฅ๐‘›โˆ’๐œ–๐‘›,๐‘ฅ๐‘›+๐œ–๐‘›)โŠ‚๐‘ˆ๎…ž๐‘›. Then (๐‘ฅ0โˆ’๐œ–0,๐‘ฅ0+๐œ–0) and all of (๐‘ฅ๐‘›โˆ’๐œ–๐‘›/2,๐‘ฅ๐‘›+๐œ–๐‘›/2), ๐‘›โˆˆโ„•, are mutually disjoint open sets. By Proposition 2.6, there are clopen neighborhoods ๐‘ˆ๐‘› of ๐‘ฅ๐‘› such that ๐‘ˆ๐‘›โŠ‚(๐‘ฅ๐‘›โˆ’๐œ–๐‘›/2,๐‘ฅ๐‘›+๐œ–๐‘›/2) and a clopen neighborhood ๐‘ˆ0 of ๐‘ฅ0 such that ๐‘ˆ0โŠ‚(๐‘ฅ0โˆ’๐œ–0/2,๐‘ฅ0+๐œ–0/2). โ‹ƒโˆž๐‘›=1๐‘ˆ๐‘› is open as it is the union of open sets, but it is also closed as we will show hereinafter. Let โ‹ƒ๐‘ฅโˆˆcl(โˆž๐‘›=1๐‘ˆ๐‘›) be given. Then there exists a sequence (๐‘ง๐‘šโ‹ƒ)โŠ‚โˆž๐‘›=1๐‘ˆ๐‘› such that ๐‘ง๐‘›โ†’๐‘ฅ. (๐‘ฅ๐‘›) has no limit points, so ๐‘ฅ is separated from (๐‘ฅ๐‘›). Therefore, there exists ๐‘โˆˆโ„• such that (๐‘ฅโˆ’๐‘‘๐‘,๐‘ฅ+๐‘‘๐‘)โˆฉ{๐‘ฅ๐‘›โˆถ๐‘›โˆˆโ„•}=โˆ…. Moreover, there exists ๐‘€โˆˆโ„• such that for every ๐‘šโ‰ฅ๐‘€, |๐‘ง๐‘šโˆ’๐‘ฅ|<(1/2)๐‘‘๐‘. It follows that |๐‘ง๐‘šโˆ’๐‘ฅ๐‘˜|โ‰ฅ||๐‘ฅ๐‘˜โˆ’๐‘ฅ|โˆ’|๐‘ง๐‘šโˆ’๐‘ฅ||>๐‘‘๐‘โˆ’(1/2)๐‘‘๐‘=(1/2)๐‘‘๐‘โ‰ซ๐‘‘๐‘˜ for all ๐‘˜>๐‘ and for all ๐‘šโ‰ฅ๐‘€. ๐‘ˆ๐‘˜โŠ‚(๐‘ฅ๐‘˜โˆ’๐‘‘๐‘˜,๐‘ฅ๐‘˜+๐‘‘๐‘˜), for each ๐‘˜โˆˆโ„•; it follows that {๐‘ง๐‘šโ‹ƒโˆถ๐‘šโ‰ฅ๐‘€}โˆฉโˆž๐‘˜=๐‘+1๐‘ˆ๐‘˜=โˆ…. That is, {๐‘ง๐‘šโ‹ƒโˆถ๐‘šโ‰ฅ๐‘€}โŠ‚๐‘๐‘˜=1๐‘ˆ๐‘˜ which is a finite union of closed sets and hence is itself closed. Since ๐‘ง๐‘šโ†’๐‘ฅ and {๐‘ง๐‘šโ‹ƒโˆถ๐‘šโ‰ฅ๐‘€}โŠ‚๐‘๐‘˜=1๐‘ˆ๐‘˜ which is closed, it follows that โ‹ƒ๐‘ฅโˆˆ๐‘๐‘˜=1๐‘ˆ๐‘˜โŠ‚โ‹ƒโˆž๐‘˜=1๐‘ˆ๐‘˜. Thus, โ‹ƒโˆž๐‘˜=1U๐‘˜โ‹ƒ=cl(โˆž๐‘˜=1๐‘ˆ๐‘˜) and hence โ‹ƒโˆž๐‘˜=1๐‘ˆ๐‘˜ is closed. Define ๐‘“ on ๐’ฉ as follows:โŽงโŽชโŽจโŽชโŽฉ๐‘“(๐‘ฅ)=1if๐‘ฅโˆˆโˆž๎š๐‘›=1๐‘ˆ๐‘›,0otherwise.(3.1) Then ๐‘“ is continuous on ๐’ฉ because โ‹ƒโˆž๐‘›=1๐‘ˆ๐‘› is clopen.

In the real case, any function that is continuous on a neighborhood is also locally uniformly continuous on that neighborhood. This property does not hold in non-Archimedean fields.

Theorem 3.4. Let ๐‘ฅ0โˆˆ๐’ฉ be given and let ฮฉ be a neighborhood of ๐‘ฅ0. Then there is a continuous function ๐‘“โˆถฮฉโ†’๐’ฉ that is not locally uniformly continuous at ๐‘ฅ0.

Proof. Let ฮฉ1=ฮฉโˆฉ(๐‘ฅ0โˆ’๐‘‘,๐‘ฅ0+๐‘‘) and apply Lemma 3.3 to ๐‘ฅ0 and ฮฉ1 to get ๐‘“1, ๐‘ˆ1, (๐‘ฅ1,๐‘›), and (๐‘ฆ1,๐‘›) which correspond to ๐‘“, ๐‘ˆ0, (๐‘ฅ๐‘›), and (๐‘ฆ๐‘›), respectively, in that lemma. Let ฮฉ2=๐‘ˆ1โˆฉ(๐‘ฅ0โˆ’๐‘‘2,๐‘ฅ0+๐‘‘2) and apply Lemma 3.3 to ๐‘ฅ0 and ฮฉ2 to get ๐‘“2, ๐‘ˆ2, (๐‘ฅ2,๐‘›), and (๐‘ฆ2,๐‘›). Continuing inductively, we can apply Lemma 3.3 to ๐‘ฅ0 and ฮฉ๐‘˜+1=๐‘ˆ๐‘˜โˆฉ(๐‘ฅ0โˆ’๐‘‘๐‘˜+1,๐‘ฅ0+๐‘‘๐‘˜+1) in order to get ๐‘ˆ๐‘˜+1, ๐‘“๐‘˜+1, (๐‘ฅ๐‘˜+1,๐‘›), and (๐‘ฆ๐‘˜+1,๐‘›). The resulting ๐‘“๐‘˜โ€™s satisfy ๐‘“๐‘˜(๐‘ฆ๐‘™,๐‘›)=0 for every ๐‘˜,๐‘™,๐‘›โˆˆโ„• and ๐‘“๐‘˜(๐‘ฅ๐‘™,๐‘›)=๐›ฟ๐‘˜,๐‘™ (the Kronecker delta) for every ๐‘˜,๐‘™,๐‘›โˆˆโ„•. Let โˆ‘๐‘“=โˆž๐‘˜=1๐‘‘๐‘˜๐‘“๐‘˜, which converges (pointwise), by Proposition 2.13, since |๐‘‘๐‘˜๐‘“๐‘˜(๐‘ฅ)|โ‰ค๐‘‘๐‘˜โ†’0 as ๐‘˜โ†’โˆž for all ๐‘ฅโˆˆฮฉ.
To show that ๐‘“ is continuous on ฮฉ, let ๐‘กโˆˆฮฉ be given. Let ๐œ–>0 in ๐’ฉ be given and let ๐‘โˆˆโ„• be such that ๐‘‘๐‘โ‰ช๐œ–. For each ๐‘›<๐‘, let ๐›ฟ๐‘›>0 be such that |๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘›(๐‘ก)|<๐‘‘๐‘ whenever ๐‘ฅโˆˆฮฉ and |๐‘ฅโˆ’๐‘ก|<๐›ฟ๐‘›, which is possible since each ๐‘“๐‘› is continuous at ๐‘ก. Let ๐›ฟ=min{๐›ฟ๐‘›โˆถ๐‘›<๐‘}. Then for all ๐‘ฅโˆˆฮฉ satisfying |๐‘ฅโˆ’๐‘ก|<๐›ฟ, we have that||||=|||||๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ก)โˆž๎“๐‘›=1๐‘‘๐‘›๎€ท๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘›๎€ธ|||||โ‰ค|||||(๐‘ก)๐‘๎“๐‘›=1๐‘‘๐‘›๎€ท๐‘“๐‘›(๐‘ฅ)โˆ’๐‘“๐‘›๎€ธ|||||+|||||(๐‘ก)โˆž๎“๐‘›=๐‘+1๐‘‘๐‘›๐‘“๐‘›|||||+|||||(๐‘ฅ)โˆž๎“๐‘›=๐‘+1๐‘‘๐‘›๐‘“๐‘›|||||โ‰ค(๐‘ก)๐‘๎“๐‘›=1๐‘‘๐‘›||๐‘‘๐‘||+|||||โˆž๎“๐‘›=๐‘+1๐‘‘๐‘›|||||+|||||โˆž๎“๐‘›=๐‘+1๐‘‘๐‘›|||||<๐‘‘๐‘+๐‘‘๐‘+๐‘‘๐‘โ‰ช๐œ–.(3.2) Thus, ๐‘“ is continuous at ๐‘ก, for all ๐‘กโˆˆฮฉ, and hence ๐‘“ is continuous on ฮฉ.
Now we show that ๐‘“ is not locally uniformly continuous at ๐‘ฅ0. Let ฮ” be a neighborhood of ๐‘ฅ0, let ๐‘€โˆˆโ„• be such that (๐‘ฅ0โˆ’๐‘‘๐‘€,๐‘ฅ0+๐‘‘๐‘€)โŠ‚ฮ”, and let ๐œ–=(1/2)๐‘‘๐‘€. So ฮฉ๐‘€โŠ‚(๐‘ฅ0โˆ’๐‘‘๐‘€,๐‘ฅ0+๐‘‘๐‘€)โŠ‚ฮ”. It follows that ๐‘ฅ๐‘€,๐‘›โˆˆฮฉ๐‘€โŠ‚ฮ” and ๐‘ฆ๐‘€,๐‘›โˆˆฮฉ๐‘€โŠ‚ฮ” are such that |๐‘ฆ๐‘€,๐‘›โˆ’๐‘ฅ๐‘€,๐‘›|โ†’0 as ๐‘›โ†’โˆž, but||๐‘“๎€ท๐‘ฆ๐‘€,๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“๐‘€,๐‘›๎€ธ||=|||||โˆž๎“๐‘˜=1๐‘‘๐‘˜๎€ท๐‘“๐‘˜๎€ท๐‘ฆ๐‘€,๐‘›๎€ธโˆ’๐‘“๐‘˜๎€ท๐‘ฅ๐‘€,๐‘›|||||๎€ธ๎€ธ=๐‘‘๐‘€>๐œ–.(3.3) Therefore ๐‘“ is not locally uniformly continuous at ๐‘ฅ0.

One can in fact show that there are continuous functions which are not locally uniformly continuous without using the property of total disconnectedness of ๐’ฉ. By using the method prescribed previously, the derivatives of the constructed functions are calculated easily, and this will prove useful when dealing with local uniform differentiability in Section 4.

Example 3.5. In the following, we will provide an explicit example of a function which is continuous but not locally uniformly continuous. Imitating the proof of Theorem 3.4, we let ๐‘ฅ๐‘š,๐‘›=๐‘›๐‘‘๐‘š,๐‘ฆ๐‘š,๐‘›=๐‘›๐‘‘๐‘š+๐‘‘๐‘›+๐‘š,๐‘ˆ๐‘š,๐‘›=๎‚ป||๐‘ฅโˆˆ๐’ฉโˆถ๐‘ฅโˆ’๐‘›๐‘‘๐‘š||โ‰ชโˆผ๐‘‘๐‘š+๐‘š+1๎‚ผ,๐‘ˆ๐‘š,๐‘œ=๎‚ป๐‘ฅโˆˆ๐’ฉโˆถ|๐‘ฅ|โ‰ชโˆผ๐‘‘๐‘š+1๎‚ผ,(3.4) and let ๎‚ป๐‘‘๐‘“(๐‘ฅ)=๐‘šif๐‘ฅโˆˆ๐‘ˆ๐‘š,๐‘›forsome๐‘š,๐‘›โˆˆโ„•,0otherwise.(3.5) Then ๐‘“ is well defined on ๐’ฉ since the ๐‘ˆ๐‘š,๐‘›โ€™s are mutually disjoint sets. We will show that ๐‘“ is continuous on ๐’ฉ but ๐‘“ is not locally uniformly continuous at 0. Let ๐‘กโˆˆ๐’ฉ be given. We will distinguish the following three cases.

Case 1 (๐‘ก<0). In this case, the function ๐‘“ is constant on the open set (โˆ’โˆž,0) containing ๐‘ก, and hence ๐‘“ is continuous at ๐‘ก.

Case 2 (๐‘ก>0). Let ๐‘€โˆˆโ„• be such that ๐‘‘๐‘€+1โ‰ช๐‘กโ‰ชโˆผ๐‘‘๐‘€. Then ๐‘กโˆˆ๐‘ˆ๐‘€โˆ’1,0 but ๐‘กโˆ‰๐‘ˆ๐‘€,0. Let ฮ”=๐‘ˆ๐‘€โˆ’1,0โงต๐‘ˆ๐‘€,0. Then ฮ” is clopen, and โ‹ƒโˆž๐‘›=1๐‘ˆ๐‘€,๐‘›โŠ‚ฮ” is also clopen. โ‹ƒ๐‘“(โˆž๐‘›=1๐‘ˆ๐‘€,๐‘›)={๐‘‘๐‘€} and โ‹ƒ๐‘“(ฮ”โงตโˆž๐‘›=1๐‘ˆ๐‘€,๐‘›)={0}. Therefore ๐‘“ is continuous on ฮ” since ๐‘“ is constant on disjoint open sets that cover ฮ”. Hence ๐‘“ is continuous at ๐‘กโˆˆฮ”.

Case 3 (๐‘ก=0). Let ๐œ–>0 in ๐’ฉ be given and let ๐‘€โˆˆโ„• be such that ๐‘‘๐‘€โ‰ช๐œ–. Let ๐›ฟ=๐‘‘๐‘€+1. Then, for |๐‘ฅ|<๐›ฟ, we have that ๐‘ฅโˆ‰๐‘ˆ๐‘˜,๐‘› for all ๐‘˜<๐‘€ and every ๐‘›โˆˆโ„•. So |๐‘“(๐‘ฅ)โˆ’๐‘“(0)|โ‰ค๐‘‘๐‘€โ‰ช๐œ–. This shows that ๐‘“ is continuous at ๐‘ก=0.

Thus, ๐‘“ is continuous at ๐‘ก for all ๐‘กโˆˆ๐’ฉ and hence ๐‘“ is continuous on ๐’ฉ. To see that ๐‘“ is not locally uniformly continuous at 0, let ฮฉ be any neighborhood of 0. Let ๐‘€โˆˆโ„• be such that (โˆ’๐‘‘๐‘€,๐‘‘๐‘€)โŠ‚ฮฉ and let ๐œ–=(1/2)๐‘‘๐‘€. Then the sequences (๐‘ฅ๐‘€,๐‘›)and(๐‘ฆ๐‘€,๐‘›) in ฮฉ defined previously are such that |๐‘ฆ๐‘€,๐‘›โˆ’๐‘ฅ๐‘€,๐‘›|=๐‘‘๐‘›+๐‘€โ†’0 as ๐‘›โ†’โˆž, but |๐‘“(๐‘ฆ๐‘€,๐‘›)โˆ’๐‘“(๐‘ฅ๐‘€,๐‘›)|=๐‘‘๐‘€>๐œ–. Thus, for every neighborhood ฮฉ of 0, there exists ๐œ–0>0 such that for each ๐›ฟ>0, there are ๐‘ฅ,๐‘ฆโˆˆฮฉ such that |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ while |๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)|โ‰ฅ๐œ–0, which shows that ๐‘“ is not locally uniformly continuous at 0.

4. Local Uniform Differentiability

Definition 4.1. Let ๐ดโŠ‚๐’ฉ, let ๐‘“โˆถ๐ดโ†’๐’ฉ, and let ๐‘ฅ0โˆˆ๐ด be given. Then we say that ๐‘“ is locally uniformly differentiable (LUD) at ๐‘ฅ0 if there is a neighborhood ฮฉ of ๐‘ฅ0 in ๐ด such that for every ๐œ–>0, there exists ๐›ฟ>0 such that |๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“โ€ฒ(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)|<๐œ–|๐‘ฆโˆ’๐‘ฅ| whenever ๐‘ฅ,๐‘ฆโˆˆฮฉ and |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ.

Definition 4.2. Let ๐ดโŠ‚๐’ฉ and let ๐‘“โˆถ๐ดโ†’๐’ฉ. Then we say that ๐‘“ is locally uniformly differentiable (LUD) on ๐ด if ๐‘“ is locally uniformly differentiable at ๐‘ฅ for all ๐‘ฅโˆˆ๐ด.

Definition 4.3. Let ๐ดโŠ‚๐’ฉ, let ๐‘“โˆถ๐ดโ†’๐’ฉ, and let ๐‘˜โˆˆโ„• be given. Then we say that ๐‘“ is LUD๐‘˜ if ๐‘˜th derivative of ๐‘“, ๐‘“(๐‘˜), exists and ๐‘“(๐‘™) is LUD on ๐ด for each ๐‘™โˆˆ{0,1,โ€ฆ,๐‘˜โˆ’1}.
The following two results follow readily from Definition 4.3.

Proposition 4.4. Let ๐ดโŠ‚๐’ฉ and let ๐‘“โˆถ๐ดโ†’๐’ฉ be LUD๐‘˜. Then ๐‘“(๐‘›) is LUD๐‘™ on ๐ด for all ๐‘™โ‰ฅ1 and ๐‘›โ‰ฅ0 satisfying ๐‘™+๐‘›โ‰ค๐‘˜.

Proposition 4.5. Let ๐‘™โˆˆโ„• be given, let ๐ดโŠ‚๐’ฉ, and let ๐‘“โˆถ๐ดโ†’๐’ฉ be such that ๐‘“ is LUD๐‘™ and ๐‘“(๐‘™) is LUD on ๐ด. Then f is LUD๐‘™+1 on ๐ด.

A noteworthy property of local uniform differentiability is that it is not inherited by the function from its derivatives, nor passed from the function onto its derivatives. Indeed, the following is an explicit example of a function whose derivative is everywhere zero, but it is not itself locally uniformly differentiable.

Example 4.6. Let ๐‘ฅ๐‘š,๐‘›, ๐‘ฆ๐‘š,๐‘›, and ๐‘ˆ๐‘š,๐‘› be as in Example 3.5. Define ๐‘”โˆถ๐’ฉโ†’๐’ฉ by ๎‚ป๐‘”(๐‘ฅ)=(๐‘›โˆ’1)2๐‘‘2๐‘šif๐‘ฅโˆˆ๐‘ˆ๐‘š,๐‘›forsome๐‘š,๐‘›โˆˆโ„•,0otherwise.(4.1) Then ๐‘” is well defined since the ๐‘ˆ๐‘š,๐‘› are mutually disjoint. Let ๐‘ฅโˆˆ๐’ฉ be given. If ๐‘ฅโˆˆ๐‘ˆ๐‘š,๐‘› for some ๐‘š,๐‘›โˆˆโ„•, then ๐‘ฅโ‰ˆ๐‘›๐‘‘๐‘š and hence |๐‘”(๐‘ฅ)|=(๐‘›โˆ’1)2๐‘‘2๐‘š<|๐‘ฅ|2. Also, if ๐‘ฅโˆ‰๐‘ˆ๐‘š,๐‘› for any ๐‘š,๐‘›โˆˆโ„•, then |๐‘”(๐‘ฅ)|=0<|๐‘ฅ|2. Therefore, |๐‘”(๐‘ฅ)|<|๐‘ฅ|2 for all ๐‘ฅโˆˆ๐’ฉ.
Note that ๐‘” is locally constant on ๐’ฉโงต{0} and hence ๐‘”โ€ฒ=0 on ๐’ฉโงต{0}. We will show that ๐‘” is differentiable at 0 with ๐‘”๎…ž(0)=0 too. So let ๐œ–>0 in ๐’ฉ be given. Let ๐›ฟ=๐œ–. Then for 0<|๐‘ฅ|<๐›ฟ, we have that|||๐‘”(๐‘ฅ)โˆ’๐‘”(0)๐‘ฅ|||=|||๐‘”(๐‘ฅ)๐‘ฅ|||<|๐‘ฅ|<๐›ฟ=๐œ–.(4.2) This shows that ๐‘” is differentiable at 0 with ๐‘”๎…ž(0)=0. Altogether, ๐‘”๎…ž=0 on ๐’ฉ. Therefore ๐‘” is ๐ถโˆž on ๐’ฉ with ๐‘”(๐‘˜)=0 for all ๐‘˜โˆˆโ„•.
Now we show that ๐‘” is not LUD at 0. Consider the sequences (๐‘ฅ๐‘›,๐‘›)๐‘›โˆˆโ„• and (๐‘ฆ๐‘›,๐‘›)๐‘›โˆˆโ„•, where ๐‘ฅ๐‘›,๐‘›=๐‘›๐‘‘๐‘› and ๐‘ฆ๐‘›,๐‘›=๐‘›๐‘‘๐‘›+๐‘‘2๐‘›. Both of these sequences converge to zero. Moreover,||๐‘ฆ๐‘›,๐‘›โˆ’๐‘ฅ๐‘›,๐‘›||=๐‘‘2๐‘›โŸถ0as๐‘›โŸถโˆž,(4.3) but ||๐‘”๎€ท๐‘ฆ๐‘›,๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘”๐‘›,๐‘›๎€ธ||=(๐‘›โˆ’1)2๐‘‘2๐‘›โ‰ฅ๐‘‘2๐‘›=||๐‘ฆ๐‘›,๐‘›โˆ’๐‘ฅ๐‘›,๐‘›||for๐‘›โ‰ฅ2.(4.4) Thus, for any neighborhood ฮฉ of 0, ๐œ–0=1, and for any ๐›ฟ>0, there are ๐‘ฆ,๐‘ฅโˆˆฮฉ such that |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ, but |๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”โ€ฒ(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)|โ‰ฅ|๐‘ฆโˆ’๐‘ฅ|. This shows that ๐‘” is not LUD at 0.

Example 4.6 shows that the property LUD is not necessarily inherited from the derivatives of the function, since ๐‘”โ€ฒ=0 is LUDโˆž. Here is another example that shows that the LUD property is not passed on from a function to its derivatives.

Example 4.7. Let ๐‘“โˆถ๐’ฉโ†’๐’ฉ be given by ๎‚ป๐‘“(๐‘ฅ)=๐‘ฅ๐‘‘๐œ†(๐‘ฅ)if๐‘ฅโ‰ 0,0if๐‘ฅ=0.(4.5) We will show that ๐‘“ is LUD on ๐’ฉ with derivative ๐‘“โ€ฒ๎‚ป๐‘‘(๐‘ฅ)=๐‘”(๐‘ฅ)=๐œ†(๐‘ฅ)if๐‘ฅโ‰ 0,0if๐‘ฅ=0,(4.6) and then we will show that ๐‘“โ€ฒ is not LUD by showing that it is not LUD at 0. So let ๐‘ฅ0โ‰ 0 be given. Let ฮฉ={๐‘ฅโˆˆ๐’ฉโˆถ๐‘ฅโˆผ๐‘ฅ0} which is an open (clopen) neighborhood of ๐‘ฅ0. Then for all ๐‘ฅ,๐‘ฆโˆˆฮฉ, we have that ||๐‘“||=||(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘ฆ๐‘‘๐œ†(๐‘ฆ)โˆ’๐‘ฅ๐‘‘๐œ†(๐‘ฅ)โˆ’๐‘‘๐œ†(๐‘ฅ)||=||(๐‘ฆโˆ’๐‘ฅ)๐‘ฆ๐‘‘๐œ†(๐‘ฅ0)โˆ’๐‘ฅ๐‘‘๐œ†(๐‘ฅ0)โˆ’๐‘‘๐œ†(๐‘ฅ0)||(๐‘ฆโˆ’๐‘ฅ)=0.(4.7) This shows that ๐‘“ is locally uniformly differentiable at ๐‘ฅ0 with derivative ๐‘“โ€ฒ(๐‘ฅ0)=๐‘”(๐‘ฅ0)=๐‘‘๐œ†(๐‘ฅ0).
For ๐‘ฅ0=0, let ๐œ–>0 in ๐’ฉ be given. Let ๐›ฟ=๐œ–๐‘‘. Then for all ๐‘ฅโ‰ ๐‘ฆ in (โˆ’๐›ฟ,๐›ฟ), we will show that||||||||๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)<๐œ–๐‘ฆโˆ’๐‘ฅ.(4.8)Case 1 (๐‘ฅ=0). Then ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ)=0 and ๐‘ฆโ‰ 0. It follows that ||๐‘“||=||๐‘“||(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)(๐‘ฆ)=๐‘‘๐œ†(๐‘ฆ)||๐‘ฆ||=๐‘‘๐œ†(๐‘ฆ)||||||||.๐‘ฆโˆ’๐‘ฅ<๐œ–๐‘ฆโˆ’๐‘ฅ(4.9) because ๐‘‘๐œ†(๐‘ฆ)โˆผ|๐‘ฆ|<๐›ฟโ‰ช๐œ–.Case 2 (๐‘ฆ=0). Then ๐‘“(๐‘ฆ)=0 and ๐‘ฅโ‰ 0. It follows that ||๐‘“||=||||=||(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)โˆ’๐‘“(๐‘ฅ)+๐‘”(๐‘ฅ)๐‘ฅโˆ’๐‘‘๐œ†(๐‘ฅ)๐‘ฅ+๐‘‘๐œ†(๐‘ฅ)๐‘ฅ||||||.=0<๐œ–๐‘ฆโˆ’๐‘ฅ(4.10)Case 3 (๐‘ฅโ‰ 0โ‰ ๐‘ฆ). Then||๐‘“||=||๐‘‘(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐œ†(๐‘ฆ)๐‘ฆโˆ’๐‘‘๐œ†(๐‘ฅ)๐‘ฅโˆ’๐‘‘๐œ†(๐‘ฅ)||=||๐‘‘(๐‘ฆโˆ’๐‘ฅ)๐œ†(๐‘ฆ)โˆ’๐‘‘๐œ†(๐‘ฅ)||||๐‘ฆ||<๐‘‘โˆ’1||||||๐‘ฆ||๐‘ฆโˆ’๐‘ฅ<๐‘‘โˆ’1๐›ฟ||||||||.๐‘ฆโˆ’๐‘ฅ=๐œ–๐‘ฆโˆ’๐‘ฅ(4.11) Thus, ๐‘“ is locally uniformly differentiable at 0 with derivative ๐‘“๎…ž(0)=๐‘”(0)=0. Altogether, it follows that ๐‘“ is LUD on ๐’ฉ with derivative ๐‘“โ€ฒ๎‚ป๐‘‘(๐‘ฅ)=๐‘”(๐‘ฅ)=๐œ†(๐‘ฅ)if๐‘ฅโ‰ 0,0if๐‘ฅ=0.(4.12)
Next we show that ๐‘“โ€ฒ is not differentiable at 0. Take the sequence (๐‘ฅ๐‘›)=๐‘‘๐‘› and the sequence (๐‘ฆ๐‘›)=2๐‘‘๐‘›. Then both sequences converge to 0. Butlim๐‘›โŸถโˆž๐‘“๎…ž๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘“๎…ž(0)๐‘ฅ๐‘›=lim๐‘›โŸถโˆž๐‘‘๐‘›๐‘‘๐‘›=1,lim๐‘›โŸถโˆž๐‘“๎…ž๎€ท๐‘ฆ๐‘›๎€ธโˆ’๐‘“๎…ž(0)๐‘ฆ๐‘›=lim๐‘›โŸถโˆž๐‘‘๐‘›2๐‘‘๐‘›=12.(4.13) If ๐‘“๎…ž(๐‘ฅ) were differentiable at 0, then both limits in (4.13) would be equal to ๐‘“๎…ž๎…ž(0). Since the two limits are different, we conclude that ๐‘“๎…ž is not differentiable at 0, and hence ๐‘“๎…ž is not LUD on ๐’ฉ.
The previous two examples illustrate that the most natural definition for LUD๐‘˜ is given in Definition 4.3.

Proposition 4.8. Let ๐‘“โˆถ๐ดโ†’๐’ฉ be LUD at ๐‘ฅ0โˆˆ๐ด. Then ๐‘“ is ๐ถ1 at ๐‘ฅ0.

Proof. Let ฮฉ be a neighborhood of ๐‘ฅ0 in ๐ด such that ๐‘“ is uniformly differentiable on ฮฉ and let ๐›ฟ0>0 be such that (๐‘ฅ0โˆ’๐›ฟ0,๐‘ฅ0+๐›ฟ0)โŠ‚ฮฉ. Let ๐œ–>0 in ๐’ฉ be given. Then there is ๐›ฟ>0, ๐›ฟโ‰ค๐›ฟ0, such that for all ๐‘ฅ,๐‘ฆโˆˆฮฉ satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ we have that |||๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“โ€ฒ|||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)2||||.๐‘ฆโˆ’๐‘ฅ(4.14) It follows that for 0<|๐‘ฅโˆ’๐‘ฅ0|<๐›ฟ, ||๐‘“๎…ž(๐‘ฅ)โˆ’๐‘“๎…ž๎€ท๐‘ฅ0๎€ธ||โ‰ค||||๐‘“๎€ท๐‘ฅ0๎€ธโˆ’๐‘“(๐‘ฅ)๐‘ฅ0โˆ’๐‘ฅโˆ’๐‘“๎…ž||||+||||๐‘“๎€ท๐‘ฅ(๐‘ฅ)(๐‘ฅ)โˆ’๐‘“0๎€ธ๐‘ฅโˆ’๐‘ฅ0โˆ’๐‘“๎…ž๎€ท๐‘ฅ0๎€ธ||||<๐œ–2+๐œ–2=๐œ–.(4.15)

Remark 4.9. Proposition 4.8 shows that the class of locally uniformly differentiable functions is a subset of the class of ๐ถ1 functions. However, this is still large enough to include all polynomial functions as Corollary 4.14 will show.

Proposition 4.10. Let ๐‘“โˆถ๐ดโ†’๐’ฉ be locally uniformly differentiable at ๐‘ฅ0โˆˆ๐ด. Then ๐‘“ is locally uniformly continuous at ๐‘ฅ0.

Proof. Let ฮ” be a neighborhood of ๐‘ฅ0 in ๐ด such that ๐‘“ is uniformly differentiable on ฮ”. By Proposition 4.8, ๐‘“๎…ž is continuous at ๐‘ฅ0. Let ฮฉโŠ‚ฮ” be a neighborhood of ๐‘ฅ0 such that for every ๐‘ฅโˆˆฮฉ, |๐‘“๎…ž(๐‘ฅ)|<1+|๐‘“๎…ž(๐‘ฅ0)|. Since ๐‘“ is uniformly differentiable on ฮ”, there exists ๐›ฟ1>0 such that for all ๐‘ฅ,๐‘ฆโˆˆฮฉโŠ‚ฮ” satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ1, we have that ||๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž||<||||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘ฆโˆ’๐‘ฅ.(4.16) Let ๐œ–>0 in ๐’ฉ be given. Let ๐›ฟ=min{๐›ฟ1,๐œ–/(2+|๐‘“๎…ž(๐‘ฅ0)|)}. Then for all ๐‘ฅ,๐‘ฆโˆˆฮฉโŠ‚ฮ” satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ, we have that ||๐‘“||<||||+||๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘ฆโˆ’๐‘ฅ๎…ž||||||<||||๎€ท||๐‘“(๐‘ฅ)๐‘ฆโˆ’๐‘ฅ๐‘ฆโˆ’๐‘ฅ2+๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ๎€ท||๐‘“<๐›ฟ2+๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ<๐œ–.(4.17)

Proposition 4.11. Let ๐ดโŠ‚๐’ฉ, let ๐‘ฅ0โˆˆ๐ด be given, let ๐›ผโˆˆ๐’ฉ be given, and let ๐‘“,๐‘”โˆถ๐ดโ†’๐’ฉ be LUD at ๐‘ฅ0. Then ๐‘“+๐›ผ๐‘” is LUD at ๐‘ฅ0, with derivative (๐‘“+๐›ผ๐‘”)โ€ฒ๎€ท๐‘ฅ0๎€ธ=๐‘“๎…ž๎€ท๐‘ฅ0๎€ธ+๐›ผ๐‘”๎…ž๎€ท๐‘ฅ0๎€ธ.(4.18)

Proof. Without loss of generality, we may assume that ๐›ผโ‰ 0. Let ๐œ–>0 in ๐’ฉ be given. Then there exists ๐›ฟ>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ,๐‘ฅ0+๐›ฟ)โŠ‚๐ด, ||๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)2||||,||๐‘ฆโˆ’๐‘ฅ๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)||||,2|๐›ผ|๐‘ฆโˆ’๐‘ฅ(4.19) whenever ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ,๐‘ฅ0+๐›ฟ). It follows that, for all ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ,๐‘ฅ0+๐›ฟ), we have that ||(๐‘“+๐›ผ๐‘”)(๐‘ฆ)โˆ’(๐‘“+๐›ผ๐‘”)(๐‘ฅ)โˆ’๐‘“๎…ž(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)โˆ’๐›ผ๐‘”๎…ž||=||๎€บ(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž๎€ป๎€บ(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)+๐›ผ๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž๎€ป||โ‰ค||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž||||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)+|๐›ผ|๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)2||||๐œ–๐‘ฆโˆ’๐‘ฅ+|๐›ผ|||||||||.2|๐›ผ|๐‘ฆโˆ’๐‘ฅ=๐œ–๐‘ฆโˆ’๐‘ฅ(4.20)

Proposition 4.12 (Chain Rule). Let ๐ด,๐ตโŠ‚๐’ฉ, let ๐‘ฅ0โˆˆ๐ด be given, and let ๐‘”โˆถ๐ดโ†’๐ต and ๐‘“โˆถ๐ตโ†’๐’ฉ be such that ๐‘” is LUD at ๐‘ฅ0 and ๐‘“ is LUD at ๐‘”(๐‘ฅ0). Then ๐‘“โˆ˜๐‘” is LUD at ๐‘ฅ0 with derivative (๐‘“โˆ˜๐‘”)โ€ฒ(๐‘ฅ0)=(๐‘“๎…žโˆ˜๐‘”(๐‘ฅ0))โ‹…๐‘”๎…ž(๐‘ฅ0).

Proof. Let ฮ” be a neighborhood of ๐‘”(๐‘ฅ0) in ๐ต such that ๐‘“ is uniformly differentiable on ฮ”, and let ฮฉ be a neighborhood of ๐‘ฅ0 in ๐ด such that ๐‘”(ฮฉ)โŠ‚ฮ” and ๐‘” is uniformly differentiable and uniformly continuous on ฮฉ (Proposition 4.10). The condition that ๐‘”(ฮฉ)โŠ‚ฮ” is always possible because ๐‘” is continuous. Since ๐‘” is continuous at ๐‘ฅ0 and ๐‘“๎…ž is continuous at ๐‘”(๐‘ฅ0) (Proposition 4.8), it follows that ๐‘“๎…žโˆ˜๐‘” is continuous at ๐‘ฅ0. So there exists ๐›ฟ1>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ1,๐‘ฅ0+๐›ฟ1)โŠ‚ฮฉ and for all ๐‘ฅโˆˆ(๐‘ฅ0โˆ’๐›ฟ1,๐‘ฅ0+๐›ฟ1) we have that ||๐‘“๎…ž||<||๐‘“โˆ˜๐‘”(๐‘ฅ)๎…ž๎€ท๐‘ฅโˆ˜๐‘”0๎€ธ||+1.(4.21) Also, since ๐‘”๎…ž is continuous at ๐‘ฅ0, there is a ๐›ฟ2>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ2,๐‘ฅ0+๐›ฟ2)โŠ‚ฮฉ, and for all ๐‘ฅโˆˆ(๐‘ฅ0โˆ’๐›ฟ2,๐‘ฅ0+๐›ฟ2), we have that ||๐‘”๎…ž||<||๐‘”(๐‘ฅ)๎…ž๎€ท๐‘ฅ0๎€ธ||+12.(4.22) Since ๐‘” is uniformly differentiable on ฮฉ, there is a ๐›ฟ3>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ3,๐‘ฅ0+๐›ฟ3)โŠ‚ฮฉ, and for all ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ3,๐‘ฅ0+๐›ฟ3) satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ3, we have that ||||<๎‚€||๐‘”๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)๎…ž||+1(๐‘ฅ)2๎‚||||โ‰ค๎€ท||๐‘”๐‘ฆโˆ’๐‘ฅ๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ||||.+1๐‘ฆโˆ’๐‘ฅ(4.23) Let ๐œ–>0 in ๐’ฉ be given. Then, since ๐‘“ is uniformly differentiable on ฮ”, there exists an ๐œ‚>0 such that whenever ๐‘ข,๐‘ฃโˆˆฮ” and |๐‘ขโˆ’๐‘ฃ|<๐œ‚, it follows that ||๐‘“(๐‘ฃ)โˆ’๐‘“(๐‘ข)โˆ’๐‘“๎…ž||<๐œ–(๐‘ข)(๐‘ฃโˆ’๐‘ข)2๎€ท||๐‘”๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ+1|๐‘ฃโˆ’๐‘ข|.(4.24) Since ๐‘” is uniformly differentiable on ฮฉ, there exists ๐›ฟ4>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ4,๐‘ฅ0+๐›ฟ4)โŠ‚ฮฉ, and for all ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ4,๐‘ฅ0+๐›ฟ4) satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ4, we have that ||๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)2๎€ท||๐‘“๎…ž๎€ท๐‘ฅโˆ˜๐‘”0๎€ธ||๎€ธ||||.+1๐‘ฆโˆ’๐‘ฅ(4.25) Finally, ๐‘” is uniformly continuous on ฮฉ. Thus, there exists ๐›ฟ5>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ5,๐‘ฅ0+๐›ฟ5)โŠ‚ฮฉ, and for all ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ5,๐‘ฅ0+๐›ฟ5) satisfying |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ5, we have that ||||๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)<๐œ‚.(4.26) Let ๎€ฝ๐›ฟ๐›ฟ=min1,๐›ฟ2,๐›ฟ3,๐›ฟ4,๐›ฟ5๎€พ.(4.27) Then (๐‘ฅ0โˆ’๐›ฟ,๐‘ฅ0+๐›ฟ)โŠ‚ฮฉ, and for all ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ/2,๐‘ฅ0+๐›ฟ/2), we have that |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ and |๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)|<๐œ‚; ๐‘”(๐‘ฆ),๐‘”(๐‘ฅ)โˆˆฮ” and hence ||๐‘“โˆ˜๐‘”(๐‘ฆ)โˆ’๐‘“โˆ˜๐‘”(๐‘ฅ)โˆ’๐‘“๎…žโˆ˜๐‘”(๐‘ฅ)๐‘”๎…ž||โ‰ค||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“โˆ˜๐‘”(๐‘ฆ)โˆ’๐‘“โˆ˜๐‘”(๐‘ฅ)โˆ’๐‘“๎…ž||+||๐‘“โˆ˜๐‘”(๐‘ฅ)(๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ))๎…žโˆ˜๐‘”(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘“๎…žโˆ˜๐‘”(๐‘ฅ)๐‘”(๐‘ฅ)โˆ’๐‘“๎…žโˆ˜๐‘”(๐‘ฅ)๐‘”๎…ž||=||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“(๐‘”(๐‘ฆ))โˆ’๐‘“(๐‘”(๐‘ฅ))โˆ’๐‘“๎…ž||+||๐‘“(๐‘”(๐‘ฅ))(๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ))๎…ž||||โˆ˜๐‘”(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)2๎€ท||๐‘”๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ||||+๐œ–+1๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)2๎€ท||๐‘“๎…ž๎€ท๐‘ฅโˆ˜๐‘”0๎€ธ||๎€ธ๎€ท||๐‘“+1๎…ž๎€ท๐‘ฅโˆ˜๐‘”0๎€ธ||๎€ธ||||<๐œ–+1๐‘ฆโˆ’๐‘ฅ2||||+๐œ–๐‘ฆโˆ’๐‘ฅ2||||||||.๐‘ฆโˆ’๐‘ฅ=๐œ–๐‘ฆโˆ’๐‘ฅ(4.28)

Proposition 4.13 (Product Rule). Let ๐ดโŠ‚๐’ฉ, let ๐‘ฅ0โˆˆ๐ด be given, and let ๐‘“,๐‘”โˆถ๐ดโ†’๐’ฉ be LUD at ๐‘ฅ0. Then ๐‘“๐‘” is LUD at ๐‘ฅ0 with (๐‘“๐‘”)โ€ฒ(๐‘ฅ0)=๐‘“๎…ž(๐‘ฅ0)๐‘”(๐‘ฅ0)+๐‘“(๐‘ฅ0)๐‘”๎…ž(๐‘ฅ0).

Proof. Let ฮฉ be a neighborhood of ๐‘ฅ0 in ๐ด such that ๐‘“ and ๐‘” are uniformly differentiable on ฮฉ and ๐‘” is uniformly continuous on ฮฉ (Proposition 4.10). Since ๐‘“, ๐‘”, and ๐‘“๎…ž are continuous at ๐‘ฅ0, there exists ๐›ฟ1>0 in ๐’ฉ such that (๐‘ฅ0โˆ’๐›ฟ1,๐‘ฅ0+๐›ฟ1)โŠ‚ฮฉ, and for all ๐‘ฅโˆˆ(๐‘ฅ0โˆ’๐›ฟ1,๐‘ฅ0+๐›ฟ1), we have that ||๐‘“||<||๐‘“๎€ท๐‘ฅ(๐‘ฅ)0๎€ธ||||๐‘”||<||๐‘”๎€ท๐‘ฅ+1,(๐‘ฅ)0๎€ธ||||๐‘“+1,๎…ž||<||๐‘“(๐‘ฅ)๎…ž๎€ท๐‘ฅ0๎€ธ||+1.(4.29) Let ๐œ–>0 in ๐’ฉ be given. Since ๐‘“ and ๐‘” are uniformly differentiable on ฮฉ, and ๐‘” is uniformly continuous on ฮฉ, there exists ๐›ฟ>0 such that whenever |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ and ๐‘ฅ,๐‘ฆโˆˆฮฉ, we have that ||๐‘”||<๐œ–(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)3๎€ท||๐‘“๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ,||+1๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)3๎€ท||๐‘”๎€ท๐‘ฅ0๎€ธ||๎€ธ||||,||+1๐‘ฆโˆ’๐‘ฅ๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๐œ–(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)3๎€ท||๐‘“๎€ท๐‘ฅ0๎€ธ||๎€ธ||||.+1๐‘ฆโˆ’๐‘ฅ(4.30) Let ๐‘ฅ,๐‘ฆโˆˆ(๐‘ฅ0โˆ’๐›ฟ1,๐‘ฅ0+๐›ฟ1) be such that |๐‘ฆโˆ’๐‘ฅ|<๐›ฟ. Then, ||๐‘“๎€ท๐‘“(๐‘ฆ)๐‘”(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)โˆ’๎…ž(๐‘ฅ)๐‘”(๐‘ฅ)+๐‘“(๐‘ฅ)๐‘”๎…ž๎€ธ||โ‰ค||๐‘“(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)(๐‘ฆ)๐‘”(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘“๎…ž||+||(๐‘ฅ)๐‘”(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)โˆ’๐‘“(๐‘ฅ)๐‘”๎…ž||โ‰ค||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘“(๐‘ฆ)๐‘”(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘“๎…ž||+||๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)(๐‘ฆโˆ’๐‘ฅ)๎…ž||+||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)(๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ))๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)๐‘”(๐‘ฅ)โˆ’๐‘“(๐‘ฅ)๐‘”๎…ž||=||||||(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๐‘”(๐‘ฆ)๐‘“(๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“๎…ž||+||๐‘“(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)๎…ž||||||||||+||||||(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)๐‘ฆโˆ’๐‘ฅ๐‘“(๐‘ฅ)๐‘”(๐‘ฆ)โˆ’๐‘”(๐‘ฅ)โˆ’๐‘”๎…ž||<๎€ท||๐‘”๎€ท๐‘ฅ(๐‘ฅ)(๐‘ฆโˆ’๐‘ฅ)0๎€ธ||๎€ธ๐œ–+13๎€ท||๐‘”๎€ท๐‘ฅ0๎€ธ||๎€ธ||||+๎€ท||๐‘“+1๐‘ฆโˆ’๐‘ฅ๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ๐œ–+13๎€ท||๐‘“๎…ž๎€ท๐‘ฅ0๎€ธ||๎€ธ||||+๎€ท||๐‘“๎€ท๐‘ฅ+1๐‘ฆโˆ’๐‘ฅ0๎€ธ||๎€ธ๐œ–+13๎€ท||๐‘“๎€ท๐‘ฅ0๎€ธ||๎€ธ||||=๎‚€๐œ–+1๐‘ฆโˆ’๐‘ฅ3+๐œ–3+๐œ–3๎‚||||||||.๐‘ฆโˆ’๐‘ฅ=๐œ–๐‘ฆโˆ’๐‘ฅ(4.31)

Since the function ๐‘“(๐‘ฅ)=๐‘ฅ is LUD on ๐’ฉ, as can be easily checked, it follows from Propositions 4.11 and 4.13 that any polynomial function is LUD on ๐’ฉ.

Corollary 4.14. Let ๐‘“โˆถ๐’ฉโ†’๐’ฉ be a polynomial function. Then ๐‘“ is LUD on ๐’ฉ.

Since the derivative of a polynomial function is again a polynomial function, we readily obtains the following result.

Corollary 4.15. Let ๐‘“โˆถ๐’ฉโ†’๐’ฉ be a polynomial function. Then ๐‘“ is LUDโˆž on ๐’ฉ.

Proposition 4.16. The function โ„Žโˆถ๐’ฉโงต{0}โ†’๐’ฉโงต{0} defined as โ„Ž(๐‘ฅ)=1/๐‘ฅ is LUDโˆž.

Proof. First we note that โ„Ž is infinitely often differentiable on ๐’ฉโงต{0}, with derivatives โ„Ž(๐‘™)(๐‘ฅ)=(โˆ’1)๐‘™๐‘™!๐‘ฅ๐‘™+1foreach๐‘™โˆˆโ„•.(4.32) Now we prove that โ„Ž is LUD on ๐’ฉโงต{0}. So let ๐‘ฅ0โˆˆ๐’ฉโงต{0} and let ๐œ–>0 in ๐’ฉ be given. Let ๎ƒฏ||ฮฉ=๐‘ฅโˆˆ๐’ฉโˆถ๐‘ฅโˆ’๐‘ฅ0||๎ƒฏ||๐‘ฅ<min0||๐‘‘,๐œ–๐‘‘||๐‘ฅ0||3.๎ƒฐ๎ƒฐ(4.33) Then ฮฉ is a neighborhood of ๐‘ฅ0 and ๐‘ฅโ‰ˆ๐‘ฅ0 for all ๐‘ฅโˆˆฮฉ. Moreover, for all ๐‘ฅ,๐‘ฆโˆˆฮฉ, we have that |||โ„Ž1(๐‘ฆ)โˆ’โ„Ž(๐‘ฅ)+๐‘ฅ2|||=||||1(๐‘ฆโˆ’๐‘ฅ)๐‘ฆโˆ’1๐‘ฅ+1๐‘ฅ2||||=||||1(๐‘ฆโˆ’๐‘ฅ)๐‘ฅ2โˆ’1||||||||=||||๐‘ฅ๐‘ฆ๐‘ฆโˆ’๐‘ฅ๐‘ฆโˆ’๐‘ฅ๐‘ฅ2||๐‘ฆ||||||||||,๐‘ฆโˆ’๐‘ฅ<๐œ–๐‘ฆโˆ’๐‘ฅ(4.34) since ๐‘ฅ2||๐‘ฆ||โ‰ˆ||๐‘ฅ0||3,||||โ‰ค||๐‘ฆโˆ’๐‘ฅ๐‘ฆโˆ’๐‘ฅ0||+||๐‘ฅโˆ’๐‘ฅ0||<2๐œ–๐‘‘||๐‘ฅ0||3โ‰ช๐œ–||๐‘ฅ0||3,(4.35) so that ||||๐‘ฆโˆ’๐‘ฅ๐‘ฅ2||๐‘ฆ||<๐œ–.(4.36) Thus, for all ๐‘ฅ0โˆˆ๐’ฉโงต{0}, โ„Ž is LUD at ๐‘ฅ0, and hence โ„Ž is LUD on ๐’ฉโงต{0}.
Applying Propositions 4.11 and 4.13, it then follows that โ„Ž(๐‘™)(๐‘ฅ)=(โˆ’1)๐‘™๐‘™!/๐‘ฅ๐‘™+1 is LUD on ๐’ฉโงต{0} for all ๐‘™โˆˆโ„•, and hence โ„Ž is LUDโˆž on ๐’ฉโงต{0}.

5. Inverse Function Theorem

The following version of the inverse function theorem for LUD functions was proven in [14].

Theorem 5.1 (inverse function theorem). Let