The main object of this paper is to prove the following theorem: Theorem. Let Sâ[a,b] Then there exists f : [a,b] â R such that f is differentiable on [a,b],fⲠâŚ
The main object of this paper is to prove the following theorem: Theorem. Let Sâ[a,b] Then there exists f : [a,b] â R such that f is differentiable on [a,b],fⲠis bounded on [a,b] and C = { x â [a,b], fⲠis continuous at x} = S if and only if S is a Gδ set dense in [a,b]. In section 5, consequences of the above theorem explore how badly discontinuous a derivative can be. Among these it will be shown that there exists a function f : [a,b] â R such that fⲠis a Dirichlet function, and it will be shown that there exists no function f : [a,b] â R such that D fⲠ= [a,b] â C fⲠis an interval, and an example of a âworst caseâ scenario is provided. The arguments stay within the realm of elementary classical analysis and are thus accessible to students who have encountered a first proof course in the subject.
It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the âŚ
It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take Ć n (x) = x n on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a,b] is, nonetheless, continuous on a dense subset of [a,b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.
Many texts in 'advanced calculus' present Darboux's Theorem (also known as the Intermediate Value Theorem for Derivatives) and the well-known example f(x) = { x2sin1/x, x = 0 /0, x âŚ
Many texts in 'advanced calculus' present Darboux's Theorem (also known as the Intermediate Value Theorem for Derivatives) and the well-known example f(x) = { x2sin1/x, x = 0 /0, x = 0 of a function with discontinuous derivative at the origin. But these texts typically fail to discuss the relationship between Darboux's result and the type of discontinuity a given derivative must have at such a point. It is no accident, for example, that the discontinuity of f'(x) = { 2xsin 1/x - cos 1/x, x = 0 /0, x = 0 at the origin is such that limx-0f'(x) does not exist. In this paper, we precisely identify such discontinuities. The arguments stay within the realm of elementary classical analysis and are thus accessible to students encountering a first proof course in the subject.
In recent articles Botsko and Gosser presented generalizations of the Fundamental Theorem of Calculus which replaced the antiderivative condition with a right antiderivative condition and an almost everywhere antiderivative condition, âŚ
In recent articles Botsko and Gosser presented generalizations of the Fundamental Theorem of Calculus which replaced the antiderivative condition with a right antiderivative condition and an almost everywhere antiderivative condition, respectively. In this paper versions of the Fundamental Theorem of Calculus are presented which replace the antiderivative condition with antiâDiniâderivative conditions and hence offer further generalizations. As in the case of the articles by Botsko and Gosser, the arguments given here stay within the realm of elementary classical analysis.
In introductory calculus, on the assumption that a realâvalued function is Riemann integrable on a compact interval and prior to having studied the Fundamental Theorem of Calculus, the student typically âŚ
In introductory calculus, on the assumption that a realâvalued function is Riemann integrable on a compact interval and prior to having studied the Fundamental Theorem of Calculus, the student typically practices computing the integral by evaluating limits of sequences of Riemann sums over equipartitions. When moving on to the first proof course in analysis, this approach, when applied to upper and lower integrals (whose existence is assured for bounded functions), can be helpful not only computationally, but also in providing proofs (alternative to those appearing in the texts) of some of the fundamental theorems on Riemann integration.
The student of âadvanced calculusâ encounters proofs that a composition of continuous functions is continuous and that a composition of differentiable functions is differentiable (the chain rule). When the question âŚ
The student of âadvanced calculusâ encounters proofs that a composition of continuous functions is continuous and that a composition of differentiable functions is differentiable (the chain rule). When the question âIs the composition of Riemann integrable functions itself Riemann integrable?â is posed, a standard counterexample is usually given (often as an exercise) and the matter is dropped. We offer here a closer analysis of the question by making certain conjectures and either proving or disproving them in a fashion consistent with the goals of the first proof course in mathematical analysis.
"On the Right-Hand Derivative of a Certain Integral Function." The American Mathematical Monthly, 98(8), pp. 751â752
"On the Right-Hand Derivative of a Certain Integral Function." The American Mathematical Monthly, 98(8), pp. 751â752
Suppose f is a realâvalued function defined on an interval I. Let R be the set of removable discontinuities of f and let J be the set of jump discontinuities âŚ
Suppose f is a realâvalued function defined on an interval I. Let R be the set of removable discontinuities of f and let J be the set of jump discontinuities of f. By showing that a set of real numbers which contains no accumulation points of itself is countable, we are able to conclude that R?J is countable.
Suppose f is a real valued function which is Riemann integrable on the interval [a, b]. Let Suppose further that x 0Îľ(a, b), that f is continuous on a deleted âŚ
Suppose f is a real valued function which is Riemann integrable on the interval [a, b]. Let Suppose further that x 0Îľ(a, b), that f is continuous on a deleted neighbourhood of x 0, but that f is discontinuous at x 0. We find that if x 0 is a removable discontinuity, then Fâ(x 0) exists but Fâ(x o)?f(x 0), and that if x 0 is a jump discontinuity then Fâ(x 0) does not exist. Finally, if x 0 is an essential discontinuity, we give examples to show that one may have Fâ(x 0) = f(x 0) or that Fâ(x 0) exists but Fâ(x 0) ? f(x 0) or that Fâ(x 0) does not exist.
Abstract If f is a realâvalued function defined on an interval I, we find that the set {x0ÎľI|f(x0â) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) âŚ
Abstract If f is a realâvalued function defined on an interval I, we find that the set {x0ÎľI|f(x0â) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) is countable. We are then able to conclude that the only set of discontinuities of f which is not necessarily countable is the set {x0ÎľI|f(x0â) does not exist (but is not infinite) and f(x0+)does not exist (but is not infinite)}.
(1989). Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain. Mathematics Magazine: Vol. 62, No. 1, pp. 43-48.
(1989). Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain. Mathematics Magazine: Vol. 62, No. 1, pp. 43-48.
We derive a Hermite interpolation formula with remainder for a function analytic on a disk. Given such a function, f, and given a preassigned matrix of nodes Z (all such âŚ
We derive a Hermite interpolation formula with remainder for a function analytic on a disk. Given such a function, f, and given a preassigned matrix of nodes Z (all such nodes being in a certain closed subdisk), we show that the sequence of Hermite interpolation polynomials for f associated with Z converges uniformily to f on the closed subdisk.
* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index
* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index
(1989). Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain. Mathematics Magazine: Vol. 62, No. 1, pp. 43-48.
(1989). Advanced Advanced Calculus: Counting the Discontinuities of a Real-Valued Function with Interval Domain. Mathematics Magazine: Vol. 62, No. 1, pp. 43-48.
"A Fundamental Theorem of Calculus that Applies to All Riemann Integrable Functions." Mathematics Magazine, 64(5), pp. 347â348
"A Fundamental Theorem of Calculus that Applies to All Riemann Integrable Functions." Mathematics Magazine, 64(5), pp. 347â348
"A Fundamental Theorem of Calculus that Applies to All Riemann Integrable Functions." Mathematics Magazine, 64(5), pp. 347â348
"A Fundamental Theorem of Calculus that Applies to All Riemann Integrable Functions." Mathematics Magazine, 64(5), pp. 347â348
Abstract If f is a realâvalued function defined on an interval I, we find that the set {x0ÎľI|f(x0â) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) âŚ
Abstract If f is a realâvalued function defined on an interval I, we find that the set {x0ÎľI|f(x0â) is infinite f(x0+) is infinite} (the set of vertical asymptotes of f) is countable. We are then able to conclude that the only set of discontinuities of f which is not necessarily countable is the set {x0ÎľI|f(x0â) does not exist (but is not infinite) and f(x0+)does not exist (but is not infinite)}.
Sets and Relations Functions Cardinality, Order Topology of the Line and Plane Topological Spaces Definitions Bases and Subbases Continuity and Topological Equivalence Metric and Normed Spaces Countability Separation Axioms Compactness âŚ
Sets and Relations Functions Cardinality, Order Topology of the Line and Plane Topological Spaces Definitions Bases and Subbases Continuity and Topological Equivalence Metric and Normed Spaces Countability Separation Axioms Compactness Product Spaces Connectedness Complete Metric Spaces Function Spaces Appendix Properties of the Real Numbers
Darboux functions Darboux functions in the first class of Baire Continuity and approximate continuity of derivatives The extreme derivatives of a function Reconstruction of the primitive The Zahorski classes The âŚ
Darboux functions Darboux functions in the first class of Baire Continuity and approximate continuity of derivatives The extreme derivatives of a function Reconstruction of the primitive The Zahorski classes The problem of characterizing derivatives Derivatives a.e. and generalizations Transformations via homeomorphisms Generalized derivatives Monotonicity Stationary and determining sets Behavior of typical continuous functions Miscellaneous topics Recent developments Bibliography Supplementary bibliography Terminology index Notational index.
It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the âŚ
It is well known that the uniform limit of a sequence of continuous real-valued functions defined on an interval I is itself continuous. However, if the convergence is pointwise, the limit function need not be continuous (take Ć n (x) = x n on [0, 1], for example). Boas has shown that the pointwise limit function of a sequence of continuous real-valued functions defined on the compact interval [a,b] is, nonetheless, continuous on a dense subset of [a,b]. In this paper, the notion of uniform convergence at a point is offered as an alternative to the Boas approach in establishing this and, consequently, other results. The arguments stay within the realm of a first proof course in classical mathematical analysis.