The Reverse of the Intermediate Value Theorem in Some Topological Spaces

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

On some properties of the conformable fractional derivative

In this paper, we prove that the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].

Introduction

The idea (or, perhaps better, the need) for this volume became clear to us when we were working on our monograph, Varieties of continua: from regions to points and back. 1 We deveoped an interest in various contemporary accounts of continuity: the prevailing Dedekind–Cantor account, smooth infinitesimal analysis (or synthetic differential geometry), and intuitionisic analysis. Each of these theories sanctions some long-standing properties that have been attributed to the continuous, at the expense of other properties so attributed. The intuitionistic theories violate the intermediate value theorem, while the Dedekind–Cantor one gives up the thesis that continua are unified wholes, and cannot be divided cleanly. The slogan is that continua are viscous, or sticky....

Approximate integration through remarkable points using the Intermediate Value Theorem

Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have called them according to the point of the interval we take. In conclusion we can say that the method developed here allows us to give new formulas of numerical integration and generalizes those that already exist.

3. Thinking continuously

Many topologists might choose to describe their subject as the study of continuity. There are continuous and discontinuous functions in our everyday routines. ‘Thinking continuously’ aims to provide a more rigorous sense of what continuity entails for real-valued functions of a real variable. It focuses on functions having a single numerical input and a single numerical output. The properties of continuous functions are considered and the boundedness theorem and intermediate value theorem are also explained.