Differentiability and continuity (video) | Khan Academy
Willie Fix. Relationship Between Differentiability and Continuity. All differentiable functions are continuous, but not all continuous functions are differentiable. Differentiability and continuity are the two fundamental concepts of differential you should consider the following relationship between these concepts. study of the relationships between robust statistics and mathematical derivatives. As an illustration of a Fréchet differentiable estimating functional an Ly .. We quote ". neighbourhoods . Lack of continuity of the influence function for.
It provides a rigorous framework for mathematical analysis in which every function between spaces is smooth i. Smooth infinitesimal analysis embodies a concept of intensive magnitude in the form of infinitesimal tangent vectors to curves. A tangent vector to a curve at a point p on it is a short straight line segment l passing through the point and pointing along the curve.
In fact we may take l actually to be an infinitesimal part of the curve.
The development of nonstandard and smooth infinitesimal analysis has breathed new life into the concept of infinitesimal, and—especially in connection with smooth infinitesimal analysis—supplied novel insights into the nature of the continuum. The Continuum and the Infinitesimal in the Ancient Period The opposition between Continuity and Discreteness played a significant role in ancient Greek philosophy.
This probably derived from the still more fundamental question concerning the One and the Many, an antithesis lying at the heart of early Greek thought see Stokes .
The Greek debate over the continuous and the discrete seems to have been ignited by the efforts of Eleatic philosophers such as Parmenides c. They were concerned to show that the divisibility of Being into parts leads to contradiction, so forcing the conclusion that the apparently diverse world is a static, changeless unity.
However in asserting the continuity of Being Parmenides is likely no more than underscoring its essential unity. Parmenides seems to be claiming that Being is more than merely continuous—that it is, in fact, a single whole, indeed an indivisible whole. The single Parmenidean existent is a continuum without parts, at once a continuum and an atom. If Parmenides was a synechist, his absolute monism precluded his being at the same time a divisionist. In support of Parmenides' doctrine of changelessness Zeno formulated his famous paradoxes of motion.
The doctrine of Atomism,[ 8 ] which seems to have arisen as an attempt at escaping the Eleatic dilemma, was first and foremost a physical theory. It was mounted by Leucippus fl. Atomism was challenged by Aristotle — B. A thoroughgoing synechist, he maintained that physical reality is a continuous plenum, and that the structure of a continuum, common to space, time and motion, is not reducible to anything else. His answer to the Eleatic problem was that continuous magnitudes are potentially divisible to infinity, in the sense that they may be divided anywhere, though they cannot be divided everywhere at the same time.
Aristotle identifies continuity and discreteness as attributes applying to the category of Quantity[ 9 ]. As examples of continuous quantities, or continua, he offers lines, planes, solids i. He also lays down definitions of a number of terms, including continuity. In effect, Aristotle defines continuity as a relation between entities rather than as an attribute appertaining to a single entity; that is to say, he does not provide an explicit definition of the concept of continuum.
By contrast no constituent parts of a discrete quantity can possess a common boundary. Aristotle sometimes recognizes infinite divisibility—the property of being divisible into parts which can themselves be further divided, the process never terminating in an indivisible—as a consequence of continuity as he characterizes the notion.
But on occasion he takes the property of infinite divisibility as defining continuity. It is this definition of continuity that figures in Aristotle's demonstration of what has come to be known as the isomorphism thesis, which asserts that either magnitude, time and motion are all continuous, or they are all discrete.
The question of whether magnitude is perpetually divisible into smaller units, or divisible only down to some atomic magnitude leads to the dilemma of divisibility see Miller a difficulty that Aristotle necessarily had to face in connection with his analysis of the continuum.
In the dilemma's first, or nihilistic horn, it is argued that, were magnitude everywhere divisible, the process of carrying out this division completely would reduce a magnitude to extensionless points, or perhaps even to nothingness. The second, or atomistic, horn starts from the assumption that magnitude is not everywhere divisible and leads to the equally unpalatable conclusion for Aristotle, at least that indivisible magnitudes must exist.
As a thoroughgoing materialist, Epicurus[ 12 ] — B. Like Leucippus and Democritus, Epicurus felt it necessary to postulate the existence of physical atoms, but to avoid Aristotle's strictures he proposed that these should not be themselves conceptually indivisible, but should contain conceptually indivisible parts.
Aristotle had shown that a continuous magnitude could not be composed of points, that is, indivisible units lacking extension, but he had not shown that an indivisible unit must necessarily lack extension. Epicurus met Aristotle's argument that a continuum could not be composed of such indivisibles by taking indivisibles to be partless units of magnitude possessing extension. In opposition to the atomists, the Stoic philosophers Zeno of Cition fl.
And, like Aristotle, they explicitly rejected any possible existence of void within the cosmos. The cosmos is pervaded by a continuous invisible substance which they called pneuma Greek: This pneuma—which was regarded as a kind of synthesis of air and fire, two of the four basic elements, the others being earth and water—was conceived as being an elastic medium through which impulses are transmitted by wave motion.
The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods The scholastic philosophers of Medieval Europe, in thrall to the massive authority of Aristotle, mostly subscribed in one form or another to the thesis, argued with great effectiveness by the Master in Book VI of the Physics, that continua cannot be composed of indivisibles. On the other hand, the avowed infinitude of the Deity of scholastic theology, which ran counter to Aristotle's thesis that the infinite existed only in a potential sense, emboldened certain of the Schoolmen to speculate that the actual infinite might be found even outside the Godhead, for instance in the assemblage of points on a continuous line.
A few scholars of the time, for example Henry of Harclay c. This incipient atomism met with a determined synechist rebuttal, initiated by John Duns Scotus c. One of these arguments is that if the diagonal and the side of a square were both composed of points, then not only would the two be commensurable in violation of Book X of Euclid, they would even be equal.
In the other, two unequal circles are constructed about a common centre, and from the supposition that the larger circle is composed of points, part of an angle is shown to be equal to the whole, in violation of Euclid's axiom V. William of Ockham c. For Ockham the principal difficulty presented by the continuous is the infinite divisibility of space, and in general, that of any continuum.
Ockham recognizes that it follows from the property of density that on arbitrarily small stretches of a line infinitely many points must lie, but resists the conclusion that lines, or indeed any continuum, consists of points. The most ambitious and systematic attempt at refuting atomism in the 14th century was mounted by Thomas Bradwardine c. The purpose of his Tractatus de Continuo c. The views on the continuum of Nicolaus Cusanus —64a champion of the actual infinite, are of considerable interest.
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.
Maret School BC Calculus / Relationship between differentiability and continuity
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. 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. 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. And so as we approach from the left, as X approaches C from the left, we actually have a situation where this expression right over here is going to approach negative infinity.
And if we approach from the right, if we approach with Xs larger than C, well, this is our X comma F of X, so we have a positive slope and then as we get closer, it gets more positive, more positive approaches positive infinity. But either way, it's not approaching a finite value.
And one side is approaching positive infinity, and the other side is approaching negative infinity.
This, the limit of this expression, is not going to exist. So once again, I'm not doing a rigorous proof here, but try to construct a discontinuous function where you will be able to find this.
It is very, very hard. And you might say, well, what about the situations where F is not even defined at C, which for sure you're not gonna be continuous if F is not defined at C. Well if F is not defined at C, then this part of the expression wouldn't even make sense, so you definitely wouldn't be differentiable. But now let's ask another thing. I've just given you good arguments for when you're not continuous, you're not going to be differentiable, but can we make another claim that if you are continuous, then you definitely will be differentiable?
Well, it turns out that there are for sure many functions, an infinite number of functions, that can be continuous at C, but not differentiable.
So for example, this could be an absolute value function. It doesn't have to be an absolute value function, but this could be Y is equal to the absolute value of X minus C. And why is this one not differentiable at C? Well, think about what's happening. Think about this expression.
Differentiability and continuity
Remember, this expression all it's doing is calculating the slope between the point X comma F of X and the point C comma F of C. So if X is, say, out here, this is X comma F of X, it's going to be calculated, so if we take the limit as X approaches C from the left, we'll be looking at this slope.
And as we get closer, we'll be looking at this slope which is actually going to be the same. In this case it would be a negative one. So as X approaches C from the left, this expression would be negative one.
But as X approaches C from the right, this expression is going to be one.