Pathological (mathematics) - Wikipedia
In effect, Aristotle defines continuity as a relation between entities as the quotations above show, he regarded as an impossibility (at that time at least). .. In the usual development of the calculus, for any differentiable. Why is it that all differentiable functions are continuous but not all continuous functions are differentiable? Learn why in this video lesson. Differentiability and continuity are the two fundamental concepts of differential you should consider the following relationship between these concepts.
Our function is defined at C, it's equal to this value, but you can see as X becomes larger than C, it just jumps down and shifts right over here. So what would happen if you were trying to find this limit? Well, remember, all this is is a slope of a line between when X is some arbitrary value, let's say it's out here, so that would be X, this would be the point X comma F of X, and then this is the point C comma F of C right over here.
So this is C comma F of C. So if you find the left side of the limit right over here, you're essentially saying okay, let's find this slope. And then let me get a little bit closer, and let's get X a little bit closer and then let's find this slope.
And then let's get X even closer than that and find this slope.Continuity and Differentiability
And in all of those cases, it would be zero. The slope is zero. So one way to think about it, the derivative or this limit as we approach from the left, seems to be approaching zero. But what about if we were to take Xs to the right? So instead of our Xs being there, what if we were to take Xs right over here? If we get X to be even closer, let's say right over here, then this would be the slope of this line.
If we get even closer, then this expression would be the slope of this line. And so as we get closer and closer to X being equal to C, we see that our slope is actually approaching negative infinity. And most importantly, it's approaching a very different value from the right. This expression is approaching a very different value from the right as it is from the left.
And so in this case, this limit up here won't exist. So we can clearly say this is not differentiable. So once again, not a proof here. I'm just getting an intuition for if something isn't continuous, it's pretty clear, at least in this case, that it's not going to be differentiable. Let's look at another case.
Let's look at a case where we have what's sometimes called a removable discontinuity or a point discontinuity. So once again, let's say we're approaching from the left. This is X, this is the point X comma F of X. Now what's interesting is where as this expression is the slope of the line connecting X comma F of X and C comma F of C, which is this point, not that point, remember we have this removable discontinuity right over here, and so this would be this expression is calculating the slope of that line.
And then if X gets even closer to C, well, then we're gonna be calculating the slope of that line. If X gets even closer to C, we're gonna be calculating the slope of that line. In his De Mente Idiotae ofhe asserts that any continuum, be it geometric, perceptual, or physical, is divisible in two senses, the one ideal, the other actual.
Cusanus's realist conception of the actual infinite is reflected in his quadrature of the circle see Boyer , p. He took the circle to be an infinilateral regular polygon, that is, a regular polygon with an infinite number of infinitesimally short sides.
By dividing it up into a correspondingly infinite number of triangles, its area, as for any regular polygon, can be computed as half the product of the apothem in this case identical with the radius of the circleand the perimeter. The idea of considering a curve as an infinilateral polygon was employed by a number of later thinkers, for instance, Kepler, Galileo and Leibniz.
The early modern period saw the spread of knowledge in Europe of ancient geometry, particularly that of Archimedes, and a loosening of the Aristotelian grip on thinking.
Indeed, tracing the development of the continuum concept during this period is tantamount to charting the rise of the calculus.
Traditionally, geometry is the branch of mathematics concerned with the continuous and arithmetic or algebra with the discrete. The infinitesimal calculus that took form in the 16th and 17th centuries, which had as its primary subject matter continuous variation, may be seen as a kind of synthesis of the continuous and the discrete, with infinitesimals bridging the gap between the two.
The widespread use of indivisibles and infinitesimals in the analysis of continuous variation by the mathematicians of the time testifies to the affirmation of a kind of mathematical atomism which, while logically questionable, made possible the spectacular mathematical advances with which the calculus is associated. It was thus to be the infinitesimal, rather than the infinite, that served as the mathematical stepping stone between the continuous and the discrete.
Johann Kepler — made abundant use of infinitesimals in his calculations. In his Nova Stereometria ofa work actually written as an aid in calculating the volumes of wine casks, he regards curves as being infinilateral polygons, and solid bodies as being made up of infinitesimal cones or infinitesimally thin discs see Baron , pp. Such uses are in keeping with Kepler's customary use of infinitesimals of the same dimension as the figures they constitute; but he also used indivisibles on occasion.
It seems to have been Kepler who first introduced the idea, which was later to become a reigning principle in geometry, of continuous change of a mathematical object, in this case, of a geometric figure.
In his Astronomiae pars Optica of Kepler notes that all the conic sections are continuously derivable from one another both through focal motion and by variation of the angle with the cone of the cutting plane.
Galileo Galilei — advocated a form of mathematical atomism in which the influence of both the Democritean atomists and the Aristotelian scholastics can be discerned. Salviati, Galileo's spokesman, maintains, contrary to Bradwardine and the Aristotelians, that continuous magnitude is made up of indivisibles, indeed an infinite number of them. When the straight line has been bent into a circle Galileo seems to take it that that the line has thereby been rendered into indivisible parts, that is, points.
But if one considers that these parts are the sides of the infinilateral polygon, they are better characterized not as indivisible points, but rather as unbendable straight lines, each at once part of and tangent to the circle[ 15 ]. The very statement of Cavalieri's principle embodies this idea: An analogous principle holds for solids.
Cavalieri's method is in essence that of reduction of dimension: For rectification a curve has, it was later realized, to be regarded as the sum, not of indivisibles, that is, points, but rather of infinitesimal straight lines, its microsegments.
But he avoided the use of infinitesimals in the determination of tangents to curves, instead developing purely algebraic methods for the purpose.
Continuity and Infinitesimals
Some of his sharpest criticism was directed at those mathematicians, such as Fermat, who used infinitesimals in the construction of tangents. As a philosopher Descartes may be broadly characterized as a synechist. His philosophical system rests on two fundamental principles: In the Meditations Descartes distinguishes mind and matter on the grounds that the corporeal, being spatially extended, is divisible, while the mental is partless.
The identification of matter and spatial extension has the consequence that matter is continuous and divisible without limit. Since extension is the sole essential property of matter and, conversely, matter always accompanies extension, matter must be ubiquitous. Descartes' space is accordingly, as it was for the Stoics, a plenum pervaded by a continuous medium. The concept of infinitesimal had arisen with problems of a geometric character and infinitesimals were originally conceived as belonging solely to the realm of continuous magnitude as opposed to that of discrete number.
But from the algebra and analytic geometry of the 16th and 17th centuries there issued the concept of infinitesimal number. The idea first appears in the work of Pierre de Fermat see Boyer  —65 on the determination of maximum and minimum extreme values, published in Fermat applied this method in determining tangents to curves and centres of gravity.
The Continuum and the Infinitesimal in the 17th and 18th Centuries Isaac Barrow[ 17 ] —77 was one of the first mathematicians to grasp the reciprocal relation between the problem of quadrature and that of finding tangents to curves—in modern parlance, between integration and differentiation. Barrow, a thoroughgoing synechist, regarded the conflict between divisionism and atomism as a live issue, and presented a number of arguments against mathematical atomism, the strongest of which is that atomism contradicts many of the basic propositions of Euclidean geometry.
Barrow conceived of continuous magnitudes as being generated by motions, and so necessarily dependent on time, a view that seems to have had a strong influence on the thinking of his illustrious pupil Isaac Newton[ 18 ] — De analysi per aequationes numero terminorum infinitas; Methodus fluxionum et serierum infinitarum; and De quadratura curvarum.
Newton's approach to the calculus rests, even more firmly than did Barrow's, on the conception of continua as being generated by motion. But Newton's exploitation of the kinematic conception went much deeper than had Barrow's. From the fact that Newton uses the letter v for the ordinate, it may be inferred that Newton is thinking of the curve as being a graph of velocity against time.
By considering the moving line, or ordinate, as the moment of the area Newton established the generality of and reciprocal relationship between the operations of differentiation and integration, a fact that Barrow had grasped but had not put to systematic use.
Newton's explicit treatment of integration as inverse differentiation was the key to the integral calculus. In the Methodus fluxionum Newton makes explicit his conception of variable quantities as generated by motions, and introduces his characteristic notation.
Maret School BC Calculus / Relationship between differentiability and continuity
He calls the quantity generated by a motion a fluent, and its rate of generation a fluxion. A quadrature is the inverse problem, that of determining the fluents when the fluxions are given. In the preface to the De quadratura curvarum he remarks that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities.
In their place he proposes to employ what he calls the method of prime and ultimate ratio. This method, in many respects an anticipation of the limit concept, receives a number of allusions in Newton's celebrated Principia mathematica philosophiae naturalis of Newton developed three approaches for his calculus, all of which he regarded as leading to equivalent results, but which varied in their degree of rigour.
The first employed infinitesimal quantities which, while not finite, are at the same time not exactly zero. Finding that these eluded precise formulation, Newton focussed instead on their ratio, which is in general a finite number.
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If this ratio is known, the infinitesimal quantities forming it may be replaced by any suitable finite magnitudes—such as velocities or fluxions—having the same ratio. This is the method of fluxions.
In this case the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and we'd expect the outside of it, after an embedding, to work the same. Yet it does not: This article does not cite any sources.
Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy)
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December Learn how and when to remove this template message Mathematicians and those in related sciences very frequently speak of whether a mathematical object—a functiona seta space of one sort or another—is "well-behaved".
The term has no fixed formal definition, and is dependent on context, mathematical interests, fashion, and taste. To ensure that an object is "well-behaved" mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but cuts down on the generality of any conclusions reached.
Concepts like non-Euclidean geometry were once considered ill-behaved, but are now common objects of study. In both pure and applied mathematics optimizationnumerical integrationor mathematical physicsfor examplewell-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.
The opposite case is usually labeled pathological. It is not unusual to have situations in which most cases in terms of cardinality or measure are pathological, but the pathological cases will not arise in practice unless constructed deliberately.
The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not.