INTERMEDIATE VALUES AND INVERSE FUNCTIONS ON Non-archimedean Fields

Continuity or even differentiability of a function on a closed interval of a non-Archimedean field are not sufficient for the function to assume all the intermediate values, a maximum, a minimum, or a unique primitive function on the interval. These problems are due to the total disconnectedness of the field in the order topology. In this paper, we show that differentiability (in the topological sense), together with some additional mild conditions, is indeed sufficient to guarantee that the function assumes all intermediate values and has a differentiable inverse function.


Introduction
Let K be a totally ordered non-Archimedean field extension of R. We introduce the following terminology. . For x, y ∈ K, we say x ∼ y if there exist n, m ∈ N such that n|x| > |y| and m|y| > |x|; for nonnegative x, y ∈ K, we say that x is infinitely smaller than y and write x y if nx < y for all n ∈ N, and we say that x is infinitely small if x 1 and x is finite if x ∼ 1; finally, we say that x is approximately equal to y and write x ≈ y if x ∼ y and |x − y| |x|. We also set λ(x) = [x], the class of x under the equivalence relation ∼.
The set H of equivalence classes under the relation ∼, which we call magnitudes, is naturally endowed with an addition via [x]+[y] = [x ·y] and an order via [x] < [y] if |y| |x| (or |x| |y|), both of which are readily checked to be well-defined. It follows that (H, +, <) is a totally ordered group, often referred to as the Hahn group or skeleton group, whose neutral element is the class of 1. The projection λ from K to H satisfies λ(x · y) = λ(x) + λ(y) and is a valuation.
The theorem of Hahn [5] provides a complete classification of any non-Archimedean extensions K of R in terms of their skeleton group H. In fact, invoking the axiom of choice it is shown that the elements of K can be written as formal power series over the group H with real coefficients, and the set of appearing "exponents" forms a well-ordered subset of H. The coefficient of the qth power in the Hahn representation of a given x will be denoted by x [q], and the number d will be defined by d [1]  From general properties of formal power series fields [9,11], it follows that if H is divisible then K is real-closed. For a general overview of the algebraic properties of formal power series fields, we refer to the comprehensive overview by Ribenboim [12] and the book by Fuchs [4]; 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 [10].
Throughout the following, N will denote any totally ordered non-Archimedean field extension of R that is complete in the order topology and whose skeleton group is Archimedean; i.e. a subgroup of R. The smallest such field is the field of the formal Laurent series whose skeleton group is Z; and the smallest such field that is also real-closed is the field R, first introduced by Levi-Civita [7,8]. In this case H = Q, and for any element x ∈ R, the set of exponents in the Hahn representation of x is a left-finite subset of Q, i.e. below any rational bound r there are only finitely many exponents. For a detailed study of the Levi-Civita field R, we refer the reader to [1,2,3,13,14,15].
In this paper, we will derive conditions under which a differentiable function assumes all intermediate values on a closed interval and has a differentiable inverse function. Previous versions of the intermediate value theorem were proved for the case of finite domain and range, and they were based on stronger smoothness criteria, namely equidifferentiability [2] and double derivate differentiability [3]. For the important class of locally analytic functions studied in detail in [15], we prove an intermediate value theorem (as well as a maximum theorem and a mean value theorem) without any requirements on the magnitude of the first derivative or the restriction of scaling into finite domains.

Review of Continuity and Differentiability
Like in any other metric space, continuity and differentiability at a point or on a domain of N are preserved under addition, multiplication and composition of functions. We also have the following useful result.
Then f is differentiable on In the following section, we study a large class of differentiable functions and show that they assume all intermediate values on a closed interval and a differentiable inverse function.

Intermediate Value Theorem and Inverse Function Theorem
First we state the following result which will be used in the proof of Theorem 3.17, and we refer the reader to [2] for its proof.
The acronym IVT in Definition 3.2 stands for Intermediate Value Theorem. As we will see in Thereom 3.17, an IVT-function on an closed interval [a, b] assumes every intermediate value between f (a) and f (b); hence the name. It follows immediately from Definition 3.2 that

Remark 3.3. It is easy to check that the property introduced in Definition 3.2 is preserved under scaling and translation. That is, if
In fact, replacing f by g, a by (a − c 2 ) /c 1 , and b by (b − c 2 ) /c 1 yields the same factor c 1 c 3 on both sides of Equation (3.1), and the same factor c 2 1 c 3 on both sides of Equation (3.2).
We show in Theorem 3.17 that if f is an IVT-function on [a, b] then f assumes every intermediate value between f (a) and f (b) and has a differentiable inverse function. The two conditions in Definition 3.2 may seem strange, but the first condition means that the function is either constant or one-to-one with slope of uniform magnitude; when restricted to R, the uniformity of the magnitude is automatic. Also, when restricted to R, the second condition means merely that the difference quotient is bounded. Moreover, the following two examples show that one of the two conditions alone will not be sufficient.
Clearly, f does not assume the value 3π/4 which lies between f (0) = 0 and f (1) = 3. Here Equation (3.1) is satisfied since but Equation (3.2) does not hold. In this example, we even have that Proof. Let n ∈ N be as in Equation (3.2). Using Equation ( Hence  Proof. Let m ∈ N be as in Lemma 3.7, and let x = y in [a, b] be given. Then Hence f is continuous on [a, b].  Proof. Let x ∈ (a, b] be given. Then g is an IVT-function on [0, 1], with λ (g (x)) = λ(x) ≥ 0 and λ (g (x)) = 0 for all x ∈ [0, 1].
Proof. That g is an IVT-function on [0, 1] follows from Remark 3.3. Now let x ∈ [0, 1] be given. Then, The following result follows immediately from Lemma 3.13 and Corollary 3.11. Now let X ∈ [0, 1] ∩ R be given. Then Thus, for all Y = X in [0, 1] ∩ R, we have that which entails that g R is differentiable (in the real sense) at X with derivative (g R ) (X) = g (X) [0] = 0, since λ (g (X)) = 0 by Lemma 3.13. Next we show that (g R ) is continuous on [0, 1] ∩ R. As in the proof of Lemma 3.10, we have that |g (y) − g (x)| ≤ 2m|y − x| for all x, y ∈ [0, 1]. In particular, Proof. Let g : [0, 1] → N be as in Lemma 3.13. We show that g is strictly increasing on [0, 1]. Let g R be as in Lemma 3.15. Then g R is continuously differentiable on [0, 1] ∩ R and (g R ) (X) = 0 for all X ∈ [0, 1] ∩ R. Thus, g R is strictly monotone on [0, 1] ∩ R. Since g R (0) = 0 < 1 = g R (1), we obtain that g R is strictly increasing on [0, 1] ∩ R. Now let x, y ∈ [0, 1] be such that x < y, and let X = x[0] and Y = y[0]. As a first case, assume that X < Y ; then g R (X) < g R (Y ). Hence where the first term is positive and real. By Corollary 3.8, we have that . As a second case, assume that X = Y . Then y − x 1, and hence since |r (x, y)| is at most finite and hence By Corollary 3.14, we have that λ (g (x) − g (X)) ≥ λ (x − X) > 0. Since g (x) ∼ 1, since g (X) ∼ 1 and since |g (x) − g (X)| 1, we obtain that From Equations (3.5) and (3.6), we obtain that g (y)−g (x) > 0. Thus, g (x) < g (y) for all x < y in [0, 1]; and hence g is strictly increasing on [0, 1]. Since and since g is strictly increasing on [0, 1], we obtain that f is strictly increasing on where l 1 and l 2 are linear functions. Hence it suffices to show that g assumes every intermediate value between g (0) = 0 and g (1) = 1. Let g R be as in Lemma 3.15, let S ∈ (0, 1) be given, and let S R = S[0]. Then S R ∈ [0, 1] ∩ R. Since g R is continuous on [0, 1] ∩ R by Lemma 3.15, there exists X ∈ [0, 1] ∩ R such that g R (X) = S R . If g(X) = S then the result of the theorem is proved; so we may assume that g(X) = S. Thus, |S − g (X)| ≤ |S − S R | + |g R (X) − g (X)| is infinitely small. Now we proceed to find x such that 0 < |x| 1, X +x ∈ [0, 1] and g (X + x) = S. Since g is differentiable on [0, 1], we have, using Corollary 3.8, that where |r (X, X + x)| is at most finite. Transforming Equation (3.7) into a fixed point problem yields where s = S − g (X), and |s| is infinitely small. Let M = {z ∈ N : λ (z) ≥ λ (s)} and let x ∈ M be given. Since |r (X, X + x)| is at most finite and since g (X) ∼ 1, we have that .