Let f = I - k be a compact vector field of class C1 on a real Hilbert space H. In the spirit of Bolzano's theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in R...
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Let f = I - k be a compact vector field of class C1 on a real Hilbert space H. In the spirit of Bolzano's theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in R2) and Kronecker (in Rk), we prove an existence result for the zeros of f in the open unit ball B of H. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction f|S of f to the boundary S of B. As an extension of this, but not as a consequence, we obtain as well an intermediate value theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical intermediate value theorem: If a half-line with extreme q ?/ f(S) intersects transversally the function f|S for only one point of S, then any value of the connected component of H\f(S) containing q is assumed by f in B. In particular, such a component is bounded.
Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical l...
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Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16-19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions.
We introduce a class of so-called very weakly locally uniformly differentiable (VWLUD) functions at a point of a general non-Archimedean ordered field extension of the real numbers, N, which is real closed and Cauchy ...
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We introduce a class of so-called very weakly locally uniformly differentiable (VWLUD) functions at a point of a general non-Archimedean ordered field extension of the real numbers, N, which is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. This new class of functions is defined by a significantly weaker criterion than that of the class of weakly locally uniformly differentiable (WLUD) functions studied in [1], which is nonetheless sufficient for a slight variation of the inverse function theorem and intermediate value theorem. Similarly, a weaker second order criterion is derived from the previously studied WLUD2 condition for twice-differentiable functions. We show that VWLUD2 functions at a point of N satisfy the mean valuetheorem in an interval around that point.
In the ongoing debate over the role of a diagram in the proof of the intermediate value theorem (IVT), Brown's [d] takes a clear position: a diagram does constitute proof of IVT. Giaquinto's [1 points out that...
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ISBN:
(纸本)9783030542498;9783030542481
In the ongoing debate over the role of a diagram in the proof of the intermediate value theorem (IVT), Brown's [d] takes a clear position: a diagram does constitute proof of IVT. Giaquinto's [1 points out that a real continuous but nowhere differentiable function lacks a curve, therefore diagrammatic evidence must be restricted to smooth functions. By applying newly-shaped concepts such as pencil-continuity and crossing x-axis to rational and real maps, f :Q Q, f : R, he comes to the conclusion that the same diagram can represent either a false or true statement, depending on the interpretation in terms of the domain of f. We analyze Brown's and Giaquinto's arguments in mathematical, philosophical and historical contexts. Our basic observation is the equivalence of IVT and the Dedekind Cut Principle. While Brown does not address the foundational issues at all, Giaquinto seeks to characterize them by the non-mathematical concept of 'desideratum'. As for philosophy, contrary to Giaquinto, we show that the diagram itself constitutes the mathematical context rather than needs an interpretation;yet, contrary to Brown, diagram for IVT does not prove anything, since it represents the axiom (completeness) of real numbers. We adopt a historical perspective to show that both Brown's and Giaquito's arguments involve concepts that take us back to the pre-Bolzano era of non-analytic proofs of IVT.
This paper offers instructional interventions designed to support undergraduate math students’ understanding of two forms of representations of Calculus concepts, mathematical language and graphs. We first discuss is...
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For a continuous function f : [a, b] R, we prove that f has a fixed point if and only if the intervals [a(0), b(0)]:= [a, b] and [a(n), b(n)]:= [a(n-1), b(n-1)] f([a(n-1), b(n-1)]) (n = 1, 2, ) are all nonempty. More ...
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For a continuous function f : [a, b] R, we prove that f has a fixed point if and only if the intervals [a(0), b(0)]:= [a, b] and [a(n), b(n)]:= [a(n-1), b(n-1)] f([a(n-1), b(n-1)]) (n = 1, 2, ) are all nonempty. More equivalent statements for the existence of fixed points of f have also been obtained and used to derive the intermediate value theorem and the nested interval property.
The paper proves an intermediate value theorem for polynomials and power series over a valued field with an additive divisible valuation group and infinite residue field. A deeper description of the behavior of the va...
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The paper proves an intermediate value theorem for polynomials and power series over a valued field with an additive divisible valuation group and infinite residue field. A deeper description of the behavior of the valuation for power series is obtained using Hensel's lemma.
In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if n = 3, and 0 = r = n -2, then there exists a Toeplitz graph of order n with decycling number r. We also prove that...
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In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if n = 3, and 0 = r = n -2, then there exists a Toeplitz graph of order n with decycling number r. We also prove that the decycling numbers of connected Cayley graphs of order n satisfy the intermediatevalue property if and only if n = 4 or 6.
As part of a larger research study, this paper describes calculus students' reasoning about the intermediate value theorem (IVT) in verbal, written, and graphical form. During interviews, students were asked to ve...
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The classical intermediate value theorem (IVT) states that if f is a continuous real-valued function on an interval [a, b] subset of R and if y is a real number strictly between f (a) and f (b), then there exists a re...
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The classical intermediate value theorem (IVT) states that if f is a continuous real-valued function on an interval [a, b] subset of R and if y is a real number strictly between f (a) and f (b), then there exists a real number x is an element of(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 not equal 0 and f (0) := 0. However, this counterexample is a bit weak as f is discontinuous only at 0. In this note, we study a class of strong counterexamples to the converse of the IVT. In particular, we present several constructions of functions f : R -> R such that f [I] = R for every nonempty open interval I of R (f [I] := {f (x) : x is an element of I}). Note that such an f clearly satisfies the conclusion of the IVT on every interval [a, b] (and then some), yet f is necessarily nowhere continuous! This leads us to a more general study of topological spaces X = (X, T) with the property that there exists a function f : X -> X such that f [O] = X for every nonvoid open set O is an element of T.
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