Academic literature on the topic 'Theorem of intermediate value (Theorem of Bolzano)'

We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.

One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.

Kumaraswamy Vela Velupillai74 presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser51 in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

Making the transition from calculus to advanced calculus/real analysis can be challenging for undergraduate students. Part of this challenge lies in the shift in the focus of student activity, from a focus on algorithms and computational techniques to activities focused around definitions, theorems, and proofs. The goal of Realistic Mathematics Education (RME) is to support students in making this transition by building on and formalizing their informal knowledge. There are a growing number of projects in this vein at the undergraduate level, in the areas of abstract algebra (TAAFU: Larsen, 2013; Larsen & Lockwood, 2013), differential equations (IO-DE: Rasmussen & Kwon, 2007), geometry (Zandieh & Rasmussen, 2010), and linear algebra (IOLA: Wawro, et al., 2012). This project represents the first steps in a similar RME-based, inquiry-oriented instructional design project aimed at advanced calculus. The results of this project are presented as three journal articles. In the first article I describe the development of a local instructional theory (LIT) for supporting the reinvention of formal conceptions of sequence convergence, the completeness property of the real numbers, and continuity of real functions. This LIT was inspired by Cauchy's proof of the Intermediate Value Theorem, and has been developed and refined using the instructional design heuristics of RME through the course of two teaching experiments. I found that a proof of the Intermediate Value Theorem was a powerful context for supporting the reinvention of a number of the core concepts of advanced calculus. The second article reports on two students' reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined and supported one another in powerful ways. The third and final article takes the findings from the two aforementioned papers and translates them for use in an advanced calculus classroom. Specifically, Cauchy's proof of the Intermediate Value Theorem is used as an inspiration and touchstone for developing some of the core concepts of advanced calculus/real analysis: namely, sequence convergence, the completeness property of the real numbers, and continuous functions. These are presented as a succession of student investigations, within the context of students developing their own formal proof of the Intermediate Value Theorem.

O Teorema do valor médio e o Teorema do valor intermédio são importantes teoremas muito usados no Cálculo integral e diferencial e não só. Neste trabalho estamos muito interessados em perceber estes teoremas, estudá-los profundamente e perceber qual o contributo dos mesmos. Para isso, visto que estão diretamente interligados com continuidade, derivadas e integrais, tivemos necessidade de ir à origem no século XVII perceber como surgiu o Cálculo pelas mãos de Isaac Newton e Gottfried Leibniz para posteriormente compreender de forma mais integral o Teorema do valor médio e o Teorema do valor intermédio que se devem, respetivamente a Joseph Louis de Lagrange e Bernard Bolzano no século XVIII, generalizados por Michel Rolle, Augustin Cauchy e Karl Weierstrass. O contexto histórico é apresentado no Capítulo 2. No Capítulo 3 analisamos os currículos do ensino secundário em Portugal e em Cabo Verde analisando a forma como os Teoremas acima referidos são lecionados e qual o grau de profundidade do seu estudo. Para perceber os Teoremas do valor médio e intermédio é necessário ter o conhecimento de limites, continuidade e derivada. Por isso, nos capítulos 5 e 6 dedicamos os primeiros tópicos especialmente a esses temas com breves revisões sobre limites, continuidade, derivadas e integrais. O Teorema do valor intermédio (Teorema de Bolzano) nos interessa muito pelo seu corolário que garante que dada uma função f contínua e dois pontos a e b do seu domínio, se f(a).f(b) < 0 então existe c no domínio de f tal que f(c) = 0. Esse corolário não só nos diz que a equação f(x) = 0 tem pelo menos uma raíz, também nos diz que tal raíz se encontra no intervalo ]a,b[. Apresentamos também um caso particular do Teorema do valor intermédio que é o Teorema de ponto fixo. Um outro teorema com fortes ligações ao Teorema do valor intermédio é o Teorema de Weierstrass que estuda os limites máximos e mínimos numa função contínua. Este mesmo teorema é usado para provar o Teorema de Rolle. O Teorema do valor médio que diz que se f é uma função contínua em [a, b] e derivável em ]a, b[ então existe um c pertencente a ]a ,b[ tal que f '(c) é igual à taxa de variação média da função em [a, b] também é estudado neste trabalho. Enfatizamos neste trabalho a importância das premissas dos teoremas, condições essas fundamentais para que se possam aplicar os teoremas. No caso do Teorema do valor médio a função deve ser contínua no intervalo [a, b] e deve ser diferenciável/derivável em ]a, b[ conceitos estudados nos capítulos 4 e 5. Não menos importante é tratado aqui o Teorema do valor médio com aplicação para integrais. Recorremos sempre ao software Geogebra para ilustrar cada um dos teoremas estudados tentando motivar as definições e as demonstrações. Também estudaremos as generalizações dos Teoremas do valor intermédio e médio.
The mean value theorem and the intermediate value theorem are very important theorems used in the integral and differential Calculus and in other domains. In this work we are very interested in analysing these theorems, in studying them deeply and in analysing their applications and contributions. For that, because they are directly interlinked with continuity, derivability and integration, we had the need to go to the origins in the XVII century to notice how the calculus appeared from Isaac Newton's and Gottfried Leibniz's and understand in a more profound way the mean value and intermediate value theorems whose main authors are Bernard Bolzano and Joseph Louis of Lagrange in the XVIII century, generalized later by Michel Rolle,Augustin Cauchy and Karl Weierstrass. The historical context is given in Chapter 2. In Chapter 3 we analyze the secondary school curricula of both Portugal and Cabo Verde, paying attention to the above theorems, namely the way and extent to which they are taught and studied. To understand the mean and intermediate value theorems it is necessary to have the knowledge of limits, continuity and derivability. Therefore, in the chapters 5 and 6 we dedicated the first topics especially to those themes with brief revisions on limits, continuity, derivability and integration. The intermediate value theorem (Bolzano's theorem) interests us specially because of its corollary which states that given a continuous function f and two points a and b in its domain, if f(a).f(b)<0, then there exists c in its domain such that f(c) = 0. Such corollary not only states that the equation f(x) = 0 has a root but that such a root is in the interval ]a, b[. We also presented a particular case of the intermediate value theorem that is the fixed point theorem. Another theorem with strong connections to the intermediate value theorem is the Weierstrass Theorem that studies the maximums and minimus in a continuous function. This same theorem is used to prove the Rolle's Theorem. The mean value theorem, which states that if f is a continuous function in [a, b] and derivable in ]a, b[ then exists c in ]a, b[ such that f '(c) is the average rate of variation of the function in [a, b]. In this work we emphasize the importance of the premisses in the theorems above. Being in such conditions is fundamental to be able to apply those theorem. In the case of the mean value theorem the function should be continuous in the interval [a, b] and it should be derivable in ]a, b [concepts studied in chapters 4 and 5. No less important we also study the mean value theorem for integrals. We always rely on the software Geogebra to illustrate the teorems studied and to try to motivate definitions and proofs. We also study generalization of the intermediate and mean value theorems.

This chapter explores the basic concepts that arise when real numbers and continuous functions are studied, particularly the limit concept and its use in proving properties of continuous functions. It gives proofs of the Bolzano–Weierstrass and Heine–Borel theorems, and the intermediate and extreme value theorems for continuous functions. Also, the chapter uses the Heine–Borel theorem to prove uniform continuity of continuous functions on closed intervals, and its consequence that any continuous function is Riemann integrable on closed intervals. In several of these proofs there is a construction by “infinite bisection,” which can be recast as an argument about binary trees. Here, the chapter uses the role of trees to construct an object—the so-called Cantor set.