1. Historical survey 2. Convergence of Sequences 2.1 Definition of Convergence 2.2. The Basic Property of Real Numbers. 2.3. Real Line 3. Continuous Functions 3.1 Continuous Functions and Their Limits 3.2 Properties of Continuous Functions. 3.2.1 The Intermediate Value Theorem 3.2.2. Maxima and Minima of Continuous Functions 3.3. The graph of a function 4. Differential Calculus 4.1 Derivative 4...

Stephen M. Walk ([email protected]) earned bachelors and masters degrees from the University of Northern Iowa and a Ph.D. from the University of Notre Dame. He has taught at St. Cloud State University in Minnesota since 1999. His mathematical interests include computability theory and many-valued logic. His non-mathematical time tends to revolve around his three children (who created the ...

Intermediate Values With the restoration of King Louis XVIII of France in 1814, one revolution had come to an end, but another was just beginning. Historians often describe the French Revolution of 1789, along with its reactions and repercussions, as the start of the modern era. For many historians of mathematics, however, the modern era began with the Bourbon Restoration and the mathematics of...

Relations of convergence of real sequences and convergence of metric spaces are investigated. An abstract intermediate value theorem for two closed sets in the range is presented. At the end, it is proven that an arc connecting the west minimal point and the east maximal point in a simple closed curve must be identical to the upper arc or lower arc of the closed curve. provide the notation and ...

The classical Intermediate Value Theorem (IVT) states that if f is a continuous real-valued function on an interval [a, b] ⊆ R and if y is a real number strictly between f(a) and f(b), then there exists a real number x ∈ (a, b) such that f(x) = y. The standard counterexample showing that the converse of the IVT is false is the function f defined on R by f(x) := sin( 1 x ) for x 6= 0 and f(0) :=...

We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval (BE) is equivalent to weak König lemma (WKL) for trees having at most two nodes at each level, and we prove that the intermediate value theorem (IVT) is equivalent to WKL for convex trees, in the framework of constructive reverse mathematics.

This paper presents the raSAT SMT solver for polynomial constraints, which aims to handle them over both reals and integers with simple unified methodologies: (1) raSAT loop for inequalities, which extends the interval constraint propagation with testing to accelerate SAT detection, and (2) a non-constructive reasoning for equations over reals, based on the generalized intermediate value theorem.

In this paper we present a new family of minimal spectrally arbitrary patterns which allow for arbitrary spectrum by using the Nilpotent-Jacobian method introduced in [3]. The novel approach here is that we use the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns.

comnode classification within membrane residue curves (m-rcms) currently hinges on lyapunov’s theorem and therefore the computation of mathematically complex eigenvalues. this paper presents an alternative criterion for the classification of nodes within m-rcms based on the total membrane flux at node compositions. this paper demonstrates that for a system exhibiting simple permeation behaviour...