Mathematics
Continuity Over an Interval
Continuity over an interval refers to a function being continuous for all values within a specific range of numbers. In mathematical terms, a function is continuous over an interval if it is continuous at every point within that interval. This concept is important for understanding the behavior of functions and their graphical representations.
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12 Key excerpts on "Continuity Over an Interval"
eBook - PDF- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
1.5 Continuity 45 Continuity on an Interval If a function f is continuous at each number in an open interval (a, b), then we say that f is continuous on (a, b). This definition applies to infinite open intervals of the form (a, +∞), (−∞, b), and (−∞, +∞). In the case where f is continuous on (−∞, +∞), we will say that f is continuous everywhere. Because Definition involves a two-sided limit, that definition does not generally apply at the endpoints of a closed interval [a, b] or at the endpoint of an interval of the form [a, b), (a, b], (−∞, b], or [a, +∞). To remedy this problem, we will agree that a function is continuous at an endpoint of an interval if its value at the endpoint is equal to the appropriate one-sided limit at that endpoint. For example, the function graphed in Figure 1.5.3 is continuous at the right endpoint of the interval [a, b] because lim x→b − f(x) = f(b) but it is not continuous at the left endpoint because lim x→a + f(x) ≠ f(a) In general, we will say a function f is continuous from the left at c if lim x→c − f(x) = f(c) and is continuous from the right at c if lim x→c + f(x) = f(c) Using this terminology, we define continuity on a closed interval as follows. x y y = f (x) a b ▴ Figure 1.5.3 Definition 1.5.2 A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: 1. f is continuous on (a, b). 2. f is continuous from the right at a. 3. f is continuous from the left at b. Modify Definition 1.5.2 appropriately so that it applies to intervals of the form [a, +∞), (−∞, b], (a, b], and [a, b). ▶ Example 2 What can you say about the continuity of the function f (x) = √ 4 − x 2 ? Solution Because the natural domain of this function is the closed interval [−2, 2], we will need to investigate the continuity of f on the open interval (−2, 2) and at the two endpoints.
eBook - ePub- Ekkehard Kopp(Author)
- 1996(Publication Date)
- Butterworth-Heinemann(Publisher)
7Continuity on Intervals
In the previous chapter we considered several examples to show that the ‘obvious’ definition of continuity is not sufficiently precise to deal with all the situations we wish to handle. In particular, we concentrated on continuity as a local property of a function; that is, the continuity or otherwise of f at the point a is determined by the behaviour of the function in a neighbourhood of a . Thus, for example, we are able to conclude that is in fact continuous throughout its domain, since it is undefined at 0 and any point a ≠ 0 has a neighbourhood which excludes 0.This example contradicts our naive idea that the graph of a continuous function ‘is in one piece’, since the domain of is not even in one piece! However, if we restrict ourselves to functions whose domain consists of one piece, i.e. is an interval , then we can again ask how accurate our intuitive idea of continuity was. And this time it will turn out to be much closer to the truth, so that our basic theorems will at first sight appear to be ‘stating the obvious’ and so be hardly worth proving – when we apply them, however, their importance and usefulness will become clear.7.1 From interval to interval
Recall that the fundamental property that makes a set I an interval is that, given any two points a, b ∈ I , all points between a and b also belong to I . So: if a ≤ c ≤ b then c ∈ I . Suppose f : I → I is given and [a, b ] ⊂ I , with Does f take on all values between a and b? In other words, will the image under f of an interval again be an interval? Our first theorem says that it will, provided that f is continuous on I . (To be a little more precise about the behaviour off at the endpoints of the interval if I = [a, b ] : we shall assume that f is continuous on the right at the left endpoint of I and continuous on the left at the right endpoint of I
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous . By contrast, if M ( t ) denotes the amount of money in a bank account at time t , then the function jumps whenever money is deposited or withdrawn, so the function M ( t ) is discontinuous. (However, if one assumes a discrete set as the domain of function M , for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.) Real-valued continuous functions Historical infinitesimal definition Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable. Definition in terms of limits Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no holes or jumps. In general, we say that the function f is continuous at some point c of its domain if, and only if, the following holds: • The limit of f ( x ) as x approaches c through domain of f does exist and is equal to f ( c ); in mathematical notation, . If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c . Thus, for example, every function whose domain is the set of all integers is continuous. We call a function continuous if and only if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset. The notation C (Ω) or C 0 (Ω) is sometimes used to denote the set of all continuous functions with domain Ω.
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
14 Calculus and ODEs y ----------+----..... ---------x Fig 2.10 The graph of a function f(x). 2.3 What is a continuous function? This book is about calculus. It is not a book on analysis. I assume then, that you would not expect from this book a detailed treatment of notions such as continuity, limit and so on, with formal definitions and rigorous proofs. That approach to the subject is very much the concern of analysis. I hope that many of my readers will wish to continue their studies into analysis, perhaps after completing this book. Certainly their needs are well catered for by a further volume in this series, and it is one of my aims to have whetted appetites for future work. At my own university, we run a one semester module in calculus in parallel with a module on mathematical reasoning, to be followed by a module on analysis. This does not mean, however, that calculus is simply analysis without the reasoning, without the rigour, and without the need for clear thinking, as it has sometimes been caricatured. I feel that I would be failing in my duty to the reader if I did not attempt to give some idea of what the notions of continuity and limit mean and why they can be important even to the student who does not intend to pursue the stricter and narrower demands of rigour and abstraction. I shall steer clear of the usual E, 8 definition of continuity, and instead present an approach which will, I hope, get over the main ideas while being mathematically correct and leading to useful applications. What is a continuous function? Let J be a function and Xo be a point of the domain of! To start with, we should be clear what it means for the functionJto be continuous at Xo. What this means, very roughly, is that if x is close to Xo thenJ(x) is close toJ(xo), and that as x approaches Xo thenj(x) will approachj'(x-). This is, however, too imprecise to qualify as a definition of continuity.
eBook - PDFNumbers and Functions
Steps into Analysis
- R. P. Burn(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
A connected set is called an interval. Bounded intervals are classified as closed intervals [ a , b ] , open intervals ( a , b ) , half-open intervals ( a , b ] and singletons { a } . Unbounded intervals are classified as closed half-rays [ a , +∞ ) or ( −∞ , a ] , open half-rays ( a , +∞ ) or ( −∞ , a ) , or the whole line R . The Intermediate Value Theorem qns 14 , 15 A continuous function f : [ a , b ] → R takes every value between f ( a ) and f ( b ) . Theorem The range of a continuous function defined on qn 21 an interval is always an interval. Theorem If a continuous function f is defined on an qns 22 , 23 , interval, f has an inverse function if and only if f 24 , 25 is strictly monotonic. In this case the inverse function is continuous and strictly monotonic. Historical Note 189 The Maximum–Minimum Theorem qns 31 , 32 , A continuous real function defined on a closed 34 interval is bounded, and attains its bounds. Uniform continuity Definition A function f : A → R is said to be uniformly qns 34 – 42 continuous on A when, given ε > 0, there exists a δ such that | x − y | < δ ⇒ | f ( x ) − f ( y ) | < ε . Theorem If a real function is continuous on a closed interval qn 43 then it is uniformly continuous on that interval. Theorem If a function is uniformly continuous on a dense qn 47 subset of a closed interval, including the end points, then it may be extended to a continuous function on the whole interval. Historical Note The Intermediate Value Theorem had been assumed from geometrical perceptions during the eighteenth century as the basis of work on approximations to roots of equations. There were mathematicians who regarded it as the essential characterisation of continuity. (We saw how wrong that was in qn 16 . The inadmissibility of defining continuous functions by the Intermediate Value Theorem was pointed out by Darboux in 1875.) In 1817 Bolzano insisted that this was a theorem which required analytical proof.
eBook - PDFCalculus
Single Variable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
x y y = f (x) a b FIGURE 1.5.4 Modify Definition 1.5.2 appropriately so that it applies to intervals of the form [a, +∞), (−∞, b], (a, b], and [a, b). Definition 1.5.2 A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: 1. f is continuous on (a, b). 2. f is continuous from the right at a. 3. f is continuous from the left at b. Example 2 What can you say about the continuity of the function f (x) = 9 − x 2 ? Solution Because the natural domain of this function is the closed interval [−3, 3], we will need to investigate the continuity of f on the open interval (−3, 3) and at the two endpoints. If c is any point in the interval (−3, 3), then it follows from Theorem 1.2.2(e) that lim x →c f (x) = lim x →c 9 − x 2 = lim x →c (9 − x 2 ) = 9 − c 2 = f (c) 42 CHAPTER 1 Limits and Continuity which proves f is continuous at each point in the interval (−3, 3). The function f is also con- tinuous at the endpoints since lim x →3 - f (x) = lim x →3 - 9 − x 2 = lim x →3 - (9 − x 2 ) = 0 = f (3) lim x →−3 + f (x) = lim x →−3 + 9 − x 2 = lim x →−3 + (9 − x 2 ) = 0 = f (−3) Thus, f is continuous on the closed interval [−3, 3] (Figure 1.5.5). – 3 – 2 – 1 3 2 1 1 2 x y f (x) = √9 – x 2 3 FIGURE 1.5.5 Some Properties of Continuous Functions The following theorem, which is a consequence of Theorem 1.2.2, will enable us to reach conclu- sions about the continuity of functions that are obtained by adding, subtracting, multiplying, and dividing continuous functions. Theorem 1.5.3 If the functions f and g are continuous at c, then (a) f + g is continuous at c. (b) f − g is continuous at c. (c) f g is continuous at c. (d ) f / g is continuous at c if g(c) = 0 and has a discontinuity at c if g(c) = 0. We will prove part (d ). The remaining proofs are similar and will be left to the exercises.
eBook - PDFCounterExamples
From Elementary Calculus to the Beginnings of Analysis
- Andrei Bourchtein, Ludmila Bourchtein(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Evidently, in this case the general and limit definitions of continuity coincide. Continuity with respect to a separate variable . A function f ( x, y ) defined on X is continuous in x at a point ( a, b ) ∈ X if the function of one variable f ( x, b ) is continuous at a . Continuity in y is defined similarly. Continuity on a set . A function f ( x, y ) defined on X is continuous on a set S ⊂ X , if f ( x, y ) is continuous at every point of S . Remark . It is worth to notice that in the case when a boundary point ( a, b ) belongs to set S , continuity on S implies continuity at ( a, b ) for the original function considered only on S . Discontinuity point . Let ( a, b ) be a limit point of the domain of f ( x, y ). If f ( x, y ) is not continuous at ( a, b ), then ( a, b ) is a point of discontinuity of f ( x, y ) (or equivalently, f ( x, y ) has a discontinuity at ( a, b )). Remark 1 . Sometimes it is required that ( a, b ) should be a point of the domain of f ( x, y ). We will not impose this restriction. Remark 2 . Classification of discontinuities is usually not considered for functions of several variables. Limits and continuity 211 Continuity. Local properties Comparative properties If f ( x, y ) and g ( x, y ) are continuous at ( a, b ) then 1) if f ( x ) ≤ g ( x ) for all ( x, y ) ∈ X in a deleted neighborhood of ( a, b ), then f ( a, b ) ≤ g ( a, b ). 2) if f ( a, b ) < g ( a, b ), then f ( x, y ) < g ( x, y ) for all ( x, y ) ∈ X in a neighbor-hood of ( a, b ).
eBook - PDFMathematical Analysis
A Concise Introduction
- Bernd S. W. Schröder, Bernd S. W. Schröder(Authors)
- 2008(Publication Date)
- Wiley-Interscience(Publisher)
Because we may need to take care of endpoints, the formal- ization of the elementary definition from calculus requires an extra item (see number 4 below). Definition 3.23 Let D 2 R be an interval of nonzero length from which at mostfinitely many points have been removed and let f : D -+ R be a function. Then f is called continuous at x iff 1. f (x) is dejned, that is, x E D, and 2. lirn f ( z ) exists, and Z-+X 3. lim f ( z ) = f ( x ) , and ZX 4. I f x is an endpoint of D, use left or right limits in 2 and 3, as appropriate. f is called continuous (on D ) iff f is continuous at every x E D. We could also define continuity for functions defined on sets for which every point of the domain is contained in an interval of nonzero length. Exercise 3-32 shows that with the present definition this idea is a bit too simple to produce a sensible result. This is not a problem, because in the early part of the text the only functions whose domains are not intervals are rational functions. For these functions the pathology of Exercise 3-32 is not an issue. Therefore we relegate all concerns regarding more complicated domains to Section 16.3. 60 3. Continuous Functions X r--------- _ _ _ IT / For theorems, we will usually work with functions that are defined on intervals, because if D is an interval from which at most finitely many points were removed, then f : D +. R is continuous at x E D iff f 11 is continuous at x, where I is a maximum-sized (with respect to containment) interval contained in D that contains x. Example 3.24 1. Constantfunctions are continuous at every x E W. 2. The function f (x) = x is continuous at every x E R. 3. Thefilmtion f ( x ) = 1x1 is continuous at every x E R. Parts 1 and 2 are trivial and part 3 follows from part 2 of Example 3.5. 0 It is useful to incorporate the definition of limits directly into a characterization of continuity.
eBook - PDF- Robert G. Bartle, Donald R. Sherbert(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
CHAPTER 5 CONTINUOUS FUNCTIONS We now begin the study of the most important class of functions that arises in real analysis: the class of continuous functions. The term ‘‘continuous’’ has been used since the time of Newton to refer to the motion of bodies or to describe an unbroken curve, but it was not made precise until the nineteenth century. Work of Bernhard Bolzano in 1817 and Augustin-Louis Cauchy in 1821 identified continuity as a very significant property of functions and proposed definitions, but since the concept is tied to that of limit, it was the careful work of Karl Weierstrass in the 1870s that brought proper understanding to the idea of continuity. We will first define the notions of continuity at a point and continuity on a set, and then show that various combinations of continuous functions give rise to continuous functions. Then in Section 5.3 we establish the fundamental properties that make continuous functions so important. For instance, we will prove that a continuous function on a closed bounded interval must attain a maximum and a minimum value. We also prove that a continuous function must take on every value intermediate to any two values it attains. These properties and others are not possessed by general functions, as various examples illustrate, and thus they distinguish continuous functions as a very special class of functions. In Section 5.4 we introduce the very important notion of uniform continuity. The distinction between continuity and uniform continuity is somewhat subtle and was not fully appreciated until the work of Weierstrass and the mathematicians of his era, but it proved to be very significant in applications. We present one application to the idea of approximating continuous functions by more elementary functions (such as polynomials). Karl Weierstrass Karl Weierstrass (¼Weierstrab) (1815–1897) was born in Westphalia, Germany.
eBook - PDF- Neil A Watson(Author)
- 1993(Publication Date)
- WSPC(Publisher)
Example. If f(x) = sin i for all x ^ 0, then f(x) does not tend to a limit as x — > 0 (as we showed in a previous example). Since / is not defined at the origin, the question of its continuity there does not arise. However, there is no value we can assign to / at 0 that will make / continuous at 0. DEFINITION. If / is defined on ]a,6[, then / is called continuous on ]a, b[ if / is continuous at c for every c G ]a, b[. In this case, the 6 in the definition may depend on c as well as on e. DEFINITION. If / is defined on [a, 6], then / is called continuous on [a, b] if / is continuous on ]a,6[ and f(x) —• f(a) as x — • a+ , /(rr) — > /(&) as a; —• 6 - . Elementary properties of continuous functions In the following theorem, the condition of continuity may refer to a point or to an interval (open or closed), so long as the same interpretation is maintained throughout each statement. Theorem 3.2. (i) The sum of two continuous functions is continuous. (ii) The product of two continuous functions is continuous. (m)The quotient of two continuous functions is continuous (at any point where the denominator is not zero). The proofs are similar to those of the corresponding results on conver-gence of sequences, and are left as exercises. Example. If n G N and f(x) = x n for all x G R , then / is continuous. We can prove this by induction on n. When n = 1, the continuity of / at Elementary properties 37 any point is immediate from the definition (with 8 = e). The induction step from n = k to n = k + 1 follows from Theorem 3.2 (ii), since x k + 1 — x.x k . Example. If n £ N and f(x) = x~ n for all x ^ 0, then / is continuous by the previous example and Theorem 3.2 (iii). Example. Any polynomial k ao + ax H h a^x is continuous, by the first example and Theorem 3.2 (i). Example. Any rational function (i.e. quotient of two polynomials) is continuous, by the previous example and Theorem 3.2 (iii).
eBook - PDF- William J. Terrell(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
4.6. Continuous Functions on an Interval 111 In the example, f is not continuous at g (1) = 1 = L . Theorem 4.5.7. Suppose g : U → R and f : V → R . If g is continuous at a ∈ U and f is continuous at g ( a ) ∈ V , then f ◦ g is continuous at a . Proof. We use the sequential characterization of continuity. Let ( x n ) be any se-quence such that x n → a as n → ∞ . Then, by continuity of g at a , g ( x n ) → g ( a ), and hence by continuity of f at g ( a ), f ( g ( x n )) → f ( g ( a )). Under the hypotheses of Theorem 4.5.7, we have f ( g ( a )) = f ( lim n →∞ g ( x n )) = lim n →∞ f ( g ( x n )) where the first equality is due to continuity of g at a and the second equality is due to continuity of f at g ( a ). There is another characterization of continuity at a point that is useful later on in describing the class of Riemann integrable functions. This characterization is just as easily described in the more general setting of functions defined on subsets of R n for any fixed n ≥ 1, so it is presented later, in Exercise 8.10.7. (For interested readers there is no harm in looking ahead and working that exercise for f : D ⊂ R → R .) Exercises. Exercise 4.5.1. Verify the continuity statements made in Example 4.5.2. Exercise 4.5.2. In Example 4.5.3, if we include the point x = 1 in the domain of f , how should f (1) be defined so as to make f continuous at 1? Exercise 4.5.3. Prove: If f : D → R is continuous on D , then | f | : D → R , defined by | f | ( x ) = | f ( x ) | , is also continuous on D . Exercise 4.5.4. Let f : D → R and suppose a ∈ D . Show that f is discontinuous at a if and only if there exists an > 0 and a sequence x k → a as k → ∞ such that for every k , | f ( x k ) − f ( a ) | ≥ . Exercise 4.5.5. Let f ( x ) = x sin(1 /x ) if x = 0, and let f (0) = 0. Show that f is continuous on [0 , 1]. 4.6. Continuous Functions on an Interval We say that a function f : D → R is continuous on D if f is continuous at each x ∈ D .
eBook - PDF- Corey M. Dunn(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
But defining f (1) = 2 would “remove” this discontinuity, and produce a continuous function on all of R : � x 2 − 1 for x = 1 , f ( x ) extends to x + 1 = x − 1 � 2 for x = 1 . The study of discontinuities is a very interesting one throughout analysis. One basic realm is in the study of discontinuities of monotone functions. Not only can it be shown that every discontinuity of a monotone function is of the first kind, but that every monotone function has only a countable number of discontinuities! A more advanced example are Baire functions: these are (continuous and some discontinuous) functions which can be approximated by continuous functions. The interested reader can find more on these topics in [8] or [26] . 5.4.5 Review of important concepts from Section 5.4 1. Continuity of functions is defined in Definition 5.35. If c is a limit point of D , Theorem 5.36 asserts that f is continuous at c if and only if lim x c f ( x ) = f ( c ). Every function is continuous at an isolated → point of its domain. 2. Since there is a close link between continuity and limits, it is easy to show that reasonable combinations–in particular, composition–of continuous functions are continuous (see Theorems 5.37 and 5.38 ). Note that these facts could be proven directly and without the use of Theorem 5.36. 3. Using our previous work, a number of familiar functions can be shown to be continuous. See the table on Page 284. 4. We briefly discuss types of discontinuities at the end of the section. Exercises for Section 5.4 1. Let f : D R . Prove that f is continuous on D if and only if the → following is true for every c ∈ D : for every neighborhood V of f ( c ), there exists a neighborhood U of c so that f ( U ∩ D ) ⊆ V . (This fact is a very important one in understanding a more general notion of continuity of functions in the broader field of topology .) 2.
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