Theorems of Continuity

6 Key excerpts on "Theorems of Continuity"

• 

  eBook - ePub

  Sets, Sequences and Mappings

  The Basic Concepts of Analysis

  • Kenneth Anderson, Dick Wick Hall(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Note the interesting pattern of growth for the collection of continuous functions. As soon as we determine a continuous function, we can combine it with other continuous functions (using the sum, product, difference, and quotient theorems) to construct additional continuous functions. We may also consider the composite function of two continuous functions, and wonder whether or not it is continuous. This question is answered affirmatively in the following theorem.

  Theorem 1.22. Let A , B, and C be sets in R 1 , let f and g be functions such that f : A → B and g : B → C , and let h be the composite function gf ; so that h: A → C. If f and g are continuous, then h is continuous.

  Proof. Let p be any point in A, and let 〈p n 〉 be a sequence of points in A such that lim p n = p. We want to show that lim h (p n ) = h (p ). Since f is continuous, we know that lim f (p n ) = f (p ); that is, 〈f (p n )〉 is a sequence of points in B which converges to the point f (p ) in B. Then, since g is continuous, we have lim g [f (p n )] = g [f (p )]. Thus, by definition of a composite function, lim gf (p n ) = gf (p ), or lim h (p n ) = h (p ).

  QED

  Now that we have determined a huge collection of continuous functions, we concern ourselves with some of the important properties of continuity. The serious student of calculus should already be aware of a few such properties, although he may not immediately recognize them here in our more general approach. Certainly, even the definition of a continuous function (as one that preserves convergent sequences) should emphasize that we are dealing with a very strong property.

  Our basic approach is to determine those properties of sets which are preserved by continuous functions. After the buildup in the preceding paragraph, one might vainly hope that all

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 8 The Concept of Continuity of a Function, and Points of Discontinuity 8.1 Introduction

  The study of calculus begins with the concept of limit introduced and discussed in Chapters 7a and 7b. Of all the many consequences of this concept, one of the most important is the concept of a continuous function . One cannot think of the subject of calculus without continuous functions , which we study now.

  The word continuous means much the same in mathematics as in everyday language. We can introduce the concept of continuity proceeding from a graphic representation of a function . A function is continuous if its graph is unbroken, i.e., free from sudden jumps or gaps .

  Suppose a function is defined on an interval I . We say that the function is continuous on the interval I , if its graph consists of one continuous curve , so that it can be drawn without lifting the pencil . There is no break in any of the graphs of continuous functions (Figure 8.1a –c ).

  Figure 8.1

  If the graph of a function is broken at any point “a ” of an interval, we say that the function is not continuous (or that it is discontinuous ) at “a ”. We give the following definition:

  Definition: A function is discontinuous at x = a, if and only if it is not continuous at x = a .

  This point “a ” is called the point of discontinuity of the function. The domain of a function plays an important role in the definition of continuity (and discontinuity) of a function. A function may be continuous on one set but discontinuous on another set. It is useful to recall the definitions of the domain of definition and the natural domain

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The How and Why of One Variable Calculus

  • Amol Sasane(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  3 Continuity Let I be an interval in R . So I is a set of the form ( a , b ) or [ a , b ] or ( −∞ , b ) , etc. Among all possible functions f : I → R , there is a ‘nice’ class of functions, namely ones that are continuous on I . What’s so nice about continuous functions? Continuous functions have properties that make them easy to work with in Calculus. For example, we will see that continuous functions possess two important properties, given by the Intermediate Value Theorem, and the Extreme Value Theorem. We will learn the statements and proofs of these in the course of this chapter. Functions that aren’t continuous may fail to possess these properties. Many bizarre functions make appearances in Calculus, and in order to avoid falling into pitfalls with simplistic thinking, we need definitions and assumptions of theorems to be stated carefully and clearly. 3.1 Definition of continuity In everyday speech, a ‘continuous’ process is one that proceeds without gaps of interruptions or sudden changes. What does it mean for a function f : R → R to be continuous? Roughly, f is said to be continuous on I if f has ‘no breaks’ at any point of I . If a break does occur in f , then this break will occur at some point of I . So we realise that in order to define continuity, we need to define what is meant by the notion of ‘ f being continuous at a point c ∈ I ’. Thus (based on this visual view of continuity), we first try to give the formal definition of the continuity of a function at a point below. Next, if a function is continuous at each point, then it will be called continuous. If a function has a break at a point c , then even if points x are close to c , the points f ( x ) do not get close to f ( c ) . See Figure 3.1. So ‘no break in f at c ’ should mean that f ( x ) stays close to f ( c ) whenever x is close to c .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

  • Andrei Bourchtein, Ludmila Bourchtein(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 Continuity General Remark . Due to the close connection between the concepts of limit and continuity, the majority of the examples of chapter 2 can be reformulated for continuous functions. A few of them are described in this chapter to ex-emplify the modifications required. Notice that all these properties are of a local character, and they appear in section 3.2 together with some additional examples of local properties. The greatest attention in this chapter is paid to global properties of functions related to continuity, because these issues were not discussed so far and they are important both in mathematics and applications. 3.1 Elements of theory Concepts Continuity at a point . A function f ( x ) defined on X is continuous at a point a ∈ X if for every ε > 0 there exists δ > 0 such that whenever x ∈ X and | x − a | < δ it follows that | f ( x ) − f ( a ) | < ε . Remark . In calculus, a non-essential simplification that f ( x ) is defined in some neighborhood of a is frequently used. In this case the continuity can be rewritten in the form lim x → a f ( x ) = f ( a ) or f ( x ) → x → a f ( a ). Actually the original definition of continuity and the definition by limit differ only in one “pathological” situation of an isolated point a of X , because the limit definition requires a to be a limit point of X , while the general definition of continuity does not contain such a requirement. It means that at any isolated point a of X , a function f ( x ) is continuous according to the general definition, but not continuous according to the definition by limit. Since the behavior of a function at an isolated point is hardly of any interest, 53 54 Counterexamples: From Calculus to the Beginnings of Analysis we will usually consider the definition of continuity by limit as a complete definition.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Therefore, we have a discontinuity at 2. Note that f (x) does approach f (2) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In a case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right. 126 The definition of the limit of a function 127 In the early development of calculus, almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating look into the exact meaning of conti- nuity. It was not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems. In particular, the work of J. B. J. Fourier (1758–1830) on the theory of heat forced mathematicians of the early 19th century to exam- ine more carefully the exact meaning of such concepts as function and continuity. Although the meaning of the word “continuous” seems intuitively clear to most people, it is not obvious how a good definition of this idea should be formulated. One popular dictionary explains continuity as follows: Continuity: Quality or state of being continuous. Continuous: Having continuity of parts. Trying to learn the meaning of continuity from these two statements alone is like trying to learn Chinese with only a Chinese dictionary. A satisfactory mathematical definition of continuity, expressed entirely in terms of properties of the real-number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789–1857). His definition, which is still used today, is most easily explained in terms of the limit concept to which we turn now. 3.2 The definition of the limit of a function Let f be a function defined in some open interval containing a point p, although we do not insist that f be defined at the point p itself.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Radical Approach to Real Analysis

  Second Edition

  • David Bressoud(Author)
  • 2006(Publication Date)
  • American Mathematical Society

    (Publisher)

  3.2.9. Use the generalized mean value theorem to prove that if f is twice differentiable, then there is some number c between x 0 and x 0 + 2 x such that f ( x 0 + 2 x ) − 2 f ( x 0 + x ) + f ( x 0 ) x 2 = f ( c ) . 3.2.10. Explain why the generalized mean value theorem implies that lim x → 0 f ( x 0 + 3 x ) − 3 f ( x 0 + 2 x ) + 3 f ( x 0 + x ) − f ( x 0 ) x 3 = f ( x 0 ) . 3.3 Continuity Continuity is such an obvious geometric phenomenon that only slowly did it dawn on mathematicians that it needed a precise definition. Well into the 19th century it was simply viewed as a descriptive term for curves that are unbroken. The preeminent calculus text of that era was S. F. Lacroix’s Trait´ e ´ el´ ementaire de calcul diff´ erentiel et de calcul int´ egral . It was first published in 1802. The sixth edition appeared in 1858. Unchanged throughout these editions was its definition of continuity: “ By the law of continuity is meant that which is observed in the description of lines by motion, and according to which the consecutive points of the same line succeed each other without any interval. ” This intuitive notion of continuity is useless when one tries to prove anything. The first appearance of the modern definition of continuity was published by Bernhard Bolzano in 1817 in the Proceedings of the Prague Scientific Society under the title Rein analytischer Beweis des Lehrsatzes dass zwieschen je zwey [ sic ] Werthen, die ein entgegengesetztes Re-sultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege . This roughly translates as Purely analytic proof of the theorem that between any two values that yield results of opposite sign there will be at least one real root of the equation . The title says it all. Bolzano was proving that any continuous function has the intermediate value property.

  Sign up to read

  Learn more about book