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Generic Continuous Functions and Other Strange Functions in Classical Real Analysis

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Generic Continuous Functions and Other Strange Functions in Classical Real Analysis Georgia State University ScholarWorks @ Georgia State University Mathematics Theses Department of Mathematics and Statistics 4-17-2008 Generic Continuous Functions and other Strange Functions in Classical Real Analysis Douglas Albert Woolley Follow this and additional works at: https://scholarworks.gsu.edu/math_theses Part of the Mathematics Commons Recommended Citation Woolley, Douglas Albert, "Generic Continuous Functions and other Strange Functions in Classical Real Analysis." Thesis, Georgia State University, 2008. https://scholarworks.gsu.edu/math_theses/44 This Thesis is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Theses by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A. Woolley Under the direction of Mihaly Bakonyi ABSTRACT In this paper we examine continuous functions which on the surface seem to defy well-known mathematical principles. Before describing these functions, we introduce the Baire Cate- gory theorem and the Cantor set, which are critical in describing some of the functions and counterexamples. We then describe generic continuous functions, which are nowhere differ- entiable and monotone on no interval, and we include an example of such a function. We then construct a more conceptually challenging function, one which is everywhere differen- tiable but monotone on no interval. We also examine the Cantor function, a nonconstant continuous function with a zero derivative almost everywhere. The final section deals with products of derivatives. INDEX WORDS: Baire, Cantor, Generic Functions, Nowhere Differentiable GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A. Woolley A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the College of Arts and Sciences Georgia State University 2008 Copyright by Douglas Albert Woolley 2008 GENERIC CONTINUOUS FUNCTIONS AND OTHER STRANGE FUNCTIONS IN CLASSICAL REAL ANALYSIS by Douglas A. Woolley Committee Chair : Mihaly Bakonyi Committee : Irme Patyi Lifeng Ding Electronic Version Approved Office of Graduate Studies College of Arts and Sciences Georgia State University May 2008 TABLE OF CONTENTS List of Figures v Section 1 Introduction 1 2 The Cantor Set 2 3 Baire’s Theorem 4 4 Generic Continuous Functions 6 5 An Everywhere Differentiable,Nowhere Monotone Function 13 6 The Cantor Function 20 7 Products of Derivatives 23 References 27 iv List of Figures Figure 6.1 20 v 1 Introduction Most continuous functions used in applications have several additional properties, such as differentiability at most points of the domain, or being monotone on certain intervals. It turns out, however, that the generic continuous function (a notion defined in Section 2) is not differentiable at any point, nor is it monotone on any subinterval of the domain. This is rather unexpected, and shows that the generic continuous function is quite different from the ones used in most applications. In this thesis we investigate this and other strange behaviors of continuous real-valued functions. We construct an everywhere differentiable, nowhere monotone function. Also presented is the Cantor function (or “Devil’s Staircase”), which is continuous, has derivative zero almost everywhere, but is not constant. Finally, we deal with functions which are derivatives of other functions. We first present an example of two such functions the product of which is not a derivative. A proof is then presented that the product of a derivative and an absolutely continuous function is always a derivative. The material in this thesis is organized as follows: Section 2 describes the Cantor set, which is of great importance in Real Analysis and plays a key role in constructing several counterexamples. Section 3 is a presentation of Baire’s Theorem, which is critical to Section 4, which shows that the generic continuous function is nowhere differentiable and is monotone on no subinterval. An example of such a function is then constructed. Section 5 describes a function which is everywhere differentiable but monotone on no subinterval. In Section 6 we present the Cantor function, while Section 7 deals with products of derivatives. 1 2 The Cantor Set In the study of Real Analysis, it is critical to understand the unusual properties of the Cantor set. Many examples and still more counterexamples make use of this set. Its construction appears in numerous texts; see for example [1], [6] and [8]. 1 2 Consider the closed interval [0,1]. Remove the open middle third ( 3, 3 ). Then remove the 1 open middle thirds from the two remaining segments. At this point the segments [1, 9 ], 2 1 2 7 8 [ 9, 3 ], [ 3, 9 ], and [ 9, 1] remain. Continue removing the open middle thirds of the remaining segments. After infinitely many steps what remains is a subset of the real numbers known as the Cantor set C . Since the Cantor set results from removing only open intervals, the complement of the Cantor set is open, therefore the Cantor set is closed. The measure of the complement is the sum of the measures of the removed intervals, ∞ 1 2 4 8 1 2n 1 1 1 + + + + ···= X = = · 3=1. 3 9 27 81 3 3n 3 1 − 2 3 n=0 3 Therefore, the Cantor set has measure zero. Despite this, the Cantor set is uncountable. To prove this, we consider the ternary representations of the numbers in the unit interval. 1 2 Since, in base three, 3 is equal to 0.1, and 3 is equal to 0.2, it is evident that, in removing the middle third of the unit interval, we actually removed all of the real numbers whose ternary decimal representation have a 1 in the first decimal place, except for 0.1 itself. 1 We can, however, choose to represent 3 as 0.02. No other number in the interval can be represented in ternary decimal representation without a 1 in the first decimal place. Similarly, 1 2 7 8 when removing the intervals ( 9, 9 ) and ( 9, 9 ), we remove all remaining real numbers whose 2 ternary representations have a 1 in the second decimal place, again with the exceptions of the 1 7 1 leftmost endpoints, 9 and 9 . As with 3 , these can be represented in ternary either with or 1 7 without using a 1; 9 as 0.01 or 0.002, and 9 as 0.21 or 0.202. Continuing, each nth step of the process removes all real numbers in the interval that cannot be expressed in ternary notation without a 1 in the nth decimal place. Therefore, the Cantor set consists of all real numbers in the interval [0, 1] that have a ternary decimal representation consisting of only 0’s and 2’s. There exists a surjective function from the Cantor set to the interval [0, 1]. It is defined by mapping each ternary number in the Cantor set to the binary number which results from replacing each 2 in the ternary number with a 1. Since the Cantor set consists of all ternary numbers consisting of only 2’s and 0’s after the decimal point, the range of this function then consists of all binary numbers in the interval. While this function is clearly surjective, it is not one-to-one; the two endpoints of each removed interval map to the same binary number. Since it is a surjective function, card C ≥ card [0, 1]. Since C ⊆ [0, 1], card C ≤ card [0, 1]. Hence, card C = card [0, 1], and therefore C is uncountable. It is easily shown that the Cantor set is nowhere dense. Since C is generated by removing only open sets from the unit interval, its complement is the union of open sets and is therefore an open set. Hence, C is closed and so C = C. Since C is a zero set, it contains no interval and therefore no interior points. Therefore the interior of C is empty, so C is nowhere dense. 3 3 Baire’s Theorem Baire’s Theorem is a fundamental result in the study of Real Analysis. Its proof appears in most textbooks on the subject. Since it is critical to Section 4, its proof is included as presented in [5] and [6]. ∞ Definition Let M be a metric space. A countable intersection G = T Gn of open dense n=1 subsets of M is called a thick subset of M. The complement of a thick set is thin (or meager). The empty set is always thin and the full space M is always thick in itself. A single open dense subset is thick and a single nowhere dense subset is thin. The Cantor set is thin in [0,1], and R is thin in R2. Theorem 3.1. (Baire’s Theorem) Every thick subset of a complete metric space M is dense in M. A nonempty, complete metric space is not thin; if M is the countable union of closed sets, then at least one has a nonempty interior. ∞ Proof. Let M be a complete metric space, M =6 ∅. Let G = T Gn be a thick subset of M, n=1 with each Gn open and dense in M. To prove that G is dense in M, it suffices to show that G ∩ A =6 ∅ for each nonempty open set A of M. Suppose there exists an open, nonempty set A such that G ∩ A = ∅. Therefore, A ⊂ Gc Thus, Gc ∩ A = A, ∞ c (\ Gn) ∩ A = A, n=1 ∞ c [(Gn ∩ A)=A, n=1 4 and therefore A is thin. Since no nonempty open subset of a complete metric space can be thin, A must be empty, a contradiction.
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