DOCSLIB.ORG

Generalized Differentiability of Continuous Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

fractal and fractional Article Generalized Differentiability of Continuous Functions Dimiter Prodanov 1,2 1 Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium; [email protected] 2 MMSDP, IICT, Bulgarian Academy of Sciences, Acad. G. Bonchev St., Block 25A, 1113 Sofia, Bulgaria Received: 19 October 2020; Accepted: 3 December 2020; Published: 10 December 2020 Abstract: Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated. Keywords: fractals; Smith–Volterra–Cantor set; Cantor functions; differentiable functions; Lipschitz (Hölder) classes MSC: 26A30; 26A16 1. Introduction From a certain perspective, a derivative can be interpreted as an idealization of the linear approximation. Moreover, the derivatives are indispensable tools for physical modelling. In classical physics, the second law of Newton tacitly assumes that the velocity is a differentiable function of time. This ensures mathematical modelling in terms of differential equations, and hence (almost everywhere) differentiable functions. Ampere even tried to prove that all functions are almost everywhere differentiable [1]. Now we know that this attempt was doomed to fail. Various non-differentiable functions were constructed in the 19th century and regarded with a mixture of wonder and horror (historical survey in [2]). For example, Charles Hermite wrote “I turn with terror and horror from this lamentable scourge of functions with no derivatives”, while Henri Poincaré denounced these functions as “an outrage against common sense”. The interest in fractal and non-differentiable functions was rekindled with the works of Mandelbrot in fractals [3]. Moreover, many physical phenomena give rise to models in terms of fractal, non-differentiable functions. Such an opposition leads to the question whether the ordinary (i.e., Cauchy) derivatives can be generalized locally in a way that can describe such irregular objects. This is certainly achievable for a class of singular functions, such as the De Rham’s function, which can be characterized locally in terms of fractional velocity [4,5]. Purely mathematically, the derivatives can be generalized in several ways. Derivatives can be defined in the usual way as limits of difference quotients on the accumulation sets of points [6], (Chaper 3, p. 105). This approach can be applied also to functions defined on perfect fractal sets as demonstrated in [7–9]. Having physical applications in mind, it is logical to restrict the functions of study to continuous ones. Such functions will be allowed to vary everywhere on the real line. So this condition makes the setting very different from the approach of Pravate, Gangal and Yang. Fractal Fract. 2020, 4, 56; doi:10.3390/fractalfract4040056 www.mdpi.com/journal/fractalfract Fractal Fract. 2020, 4, 56 2 of 17 A very broad generalization of derivatives can be achieved if the linear, Lipschitz, growth condition is replaced by a non-linear one. From this perspective, the growth condition can be taken as the modulus of continuity. Such notion is by necessity limited. However, it provides a framework which allows one to identify if a certain property of derivatives carries over to more particular constructs, for example in the setting of fractional velocities [10], or it is special for the linear growth condition. The paper is organized as follows. Section2 introduces some general definitions and notations. Section3 introduces the oscillation of functions. Section4 introduces indicial derivatives. Section5 gives the construction of a singular function used to clarify the Lebesgue differentiation theorem. Section6 characterizes the continuity sets of derivatives. Section7 characterizes the fractional velocities in terms of the indicator derivatives. 2. General Definitions and Conventions The term variable denotes an indefinite number taken from a predefined set, usually the real numbers. Sets are denoted by capital letters, while variables taking values in sets are denoted by lowercase. The term function denotes a mapping from one number to another and the action of the function is denoted as f (x) = y. Implicitly the mapping acts on the real numbers: f : R 7! R. If a statement of a function f fulfils a certain predicate with argument A (i.e., Pred[A]) the following short-hand notation will be used f =∼ Pred[A]. The co-domain of the function f : X 7! Y is denoted as f [X] = Y. The term operator denotes the mapping from a functional expression to functional expression. The term functional denotes the mapping from a functional expression to a number. Square brackets are used for the arguments of operators and functionals, while round brackets are used for the arguments of functions. The term Cauchy sequence will always be interpreted as a null sequence. Everywhere, e will be considered as a small positive variable. The asymptotic small O notation will be used as introduced in [5]. Definition 1. Define the parametrized difference operators acting on the function f (x) as + De [ f ](x) := f (x + e) − f (x), − De [ f ](x) := f (x) − f (x − e) for the variable e > 0. The two operators are referred to as forward difference and backward difference operators, respectively. Definition 2 (Anonymous function notation). The notation for the pair m :: e will be interpreted as the implication that if Left-Hand Side (LHS) is fixed then the Right-Hand Side (RHS) is fixed by the value chosen on the left, i.e., as an anonymous functional dependency e = e(m). 3. The Discontinuity Sets of Functions In order to state the main result of the section we introduce the auxiliary concept of function oscillation. Definition 3 (Oscillation). Define the oscillation of the function f in the interval J = [a, b] as osc f := sup f − inf f J J J Fractal Fract. 2020, 4, 56 3 of 17 Definition 4 (Directed Oscillation). Define the directed oscillations as (i) the forward oscillation: + osce [ f ] (x) := sup f − inf f , I = [x, x + e] [x,x+e] [x,x+e] and the backward oscillation: − osce [ f ] (x) := sup f − inf f , I = [x − e, x] [x−e,x] [x−e,x] Finally, define the limits, if such exist as finite numbers, as osc+[ f ](x) := lim sup f − inf f , I = [x, x + e] jIj!0 I I osc−[ f ](x) := lim sup f − inf f , I = [x − e, x] jIj!0 I I where j · j denotes the length of the interval, according to previously introduced notation [5,11]. The lemma was stated in [11]: Lemma 1 (Oscillation lemma). Consider the function f : X 7! Y ⊆ R . Suppose that I+ = [x, x + e] ⊆ X, I− = [x − e, x] ⊆ X, respectively. If osc+[ f ](x) = 0 then f is right-continuous at x. Conversely, if f is right-continuous at x then osc+[ f ](x) = 0. If osc−[ f ](x) = 0 then f is left-continuous at x. Conversely, if f is left-continuous at x then osc−[ f ](x) = 0. That is, ± lim osce [ f ] (x) = 0 () lim f (x ± e) = f (x) e!0 e!0 Then the negation of the statement is also true. Corollary 1. The following two statements are equivalent ± lim osce [ f ] (x) > 0 () lim f (x ± e) 6= f (x) e!0 e!0 Theorem 1 (Disconnected discontinuity set). Consider a bounded function F defined on the compact interval I. Then its set of discontinuity D[F, I] is totally disconnected in I. Proof. Consider a decreasing collection of closed nested intervals from a partition of the interval n I1 = [x, x + h] (h > 0, not necessarily small); that is fIk = [x, x + ak]gk=1, Ik+1 ⊂ Ik ⊂ ... ⊂ I1. Since the case when the function is locally constant is trivial we consider only two cases: increasing and decreasing. Let En = In \ In−1 and define Dn := sup F − inf F I In n Dn−1 := sup F − inf F I − In−1 n 1 DE := sup F − inf F E En n Increasing case: Suppose that F is increasing in In−1 then Fractal Fract. 2020, 4, 56 4 of 17 Dn−1 − Dn = sup F − inf F − sup F + inf F = sup F − inf F − inf F + inf F = DE = osc f I − In I − En I − En In−1 n 1 In En n 1 n 1 Decreasing case: Suppose that F is decreasing in In−1 then Dn−1 − Dn = sup F − inf F − sup F + sup F = DE = osc f E E In n In En n On the other hand, osc f = Dn and osc f = Dn−1. If F is continuous on the opening of In−1, In In−1 in limit lim osc f − osc f = 0 ) lim osc f = 0 n!¥ In In−1 jEnj!0 En and D[F, I] n fxg = Æ. On the other hand, F can be discontinuous on the boundary points fx, x + an−1g, which are disconnected. Suppose then that both endpoints are points of discontinuity. Hence, fx, x + an−1g ⊆ D[F, I]. If F is discontinuous on the opening of In−1, we take In and proceed in the same way. Therefore, D[F, I] 6= Æ is a union of totally disconnected sets and hence it is totally disconnected. The disconnected discontinuity theorem is used as the main tool characterizing the continuity sets of derivatives.
Recommended publications
  • On Semicontinuous Functions and Baire Functions

    On Semicontinuous Functions and Baire Functions

    ON SEMICONTINUOUS FUNCTIONS AND BAIRE FUNCTIONS BY ROBERT E. ZINK(») 1. According to a classical theorem, every semicontinuous real-valued func- tion of a real variable can be obtained as the limit of a sequence of continuous functions. The corresponding proposition need not hold for the semicontinuous functions defined on an arbitrary topological space; indeed, it is known that the theorem holds in the general case if and only if the topology is perfectly normal [6]. For some purposes, it is important to know when a topological measure space has the property that each semicontinuous real-valued function defined thereon is almost everywhere equal to a function of the first Baire class. In the present article, we are able to resolve this question when the topology is completely regular. A most interesting by-product of our consideration of this problem is the discovery that every Lebesgue measurable function is equivalent to the limit of a sequence of approximately continuous ones. This fact enables us to answer a question, posed in an earlier work [7], concerning the existence of topological measure spaces in which every bounded measurable function is equivalent to a Baire function of the first class. 2. We begin the discussion with some pertinent background material. The Blumberg upper measurable boundary, uB, of a real-valued function / defined on Euclidean q-space, Eq, is specified in the following manner: If Ey = {x: fix) > y}, then uBi£) = inf {y: exterior metric density ofEy at £ is zero}. The lower measurable boundary, lB, is similarly defined. Blumberg proved that uB is approximately upper-semicontinuous and that lB is approximately lower-semicontinuous and then observed that approximately semicontinuous functions are Lebesgue measurable.
  • The Baire Order of the Functions Continuous Almost Everywhere R

    The Baire Order of the Functions Continuous Almost Everywhere R

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 2, December 1973 THE BAIRE ORDER OF THE FUNCTIONS CONTINUOUS ALMOST EVERYWHERE R. d. MAULDIN Abstract. Let í> be the family of all real-valued functions defined on the unit interval / which are continuous except for a set of Lebesgue measure zero. Let <J>0be <I>and for each ordinal a, let <S>abe the family of all pointwise limits of sequences taken from {Jy<x ÍV Then <Dm is the Baire family generated by O. It is proven here that if 0<a<co1, then O^i^ . The proof is based upon the construction of a Borel measurable function h from / onto the Hubert cube Q such that if x is in Q, then h^ix) is not a subset of an F„ set of Lebesgue measure zero. If O is a family of real-valued functions defined on a set S, then the Baire family generated by O may be described as follows: Let O0=O and for each ordinal <x>0, let Oa be the family of all pointwise limits of sequences taken from \Jy<x î>r Of course, ^o^^V+i* where mx denotes the first uncountable ordinal and 4> is the Baire family generated by O; the family O^, is the smallest subfamily of Rs containing O and which is closed under pointwise limits of sequences. The order of O is the first ordinal a such that 0>X=Q>X+X. Let C denote the family of all real-valued continuous functions on the unit interval I.
  • On the Structures of Generating Iterated Function Systems of Cantor Sets

    On the Structures of Generating Iterated Function Systems of Cantor Sets

    ON THE STRUCTURES OF GENERATING ITERATED FUNCTION SYSTEMS OF CANTOR SETS DE-JUN FENG AND YANG WANG Abstract. A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimH F < 1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study. 1. Introduction N d In this paper, a family of contractive affine maps Φ = fφjgj=1 in R is called an iterated function system (IFS). According to Hutchinson [12], there is a unique non-empty compact d SN F = FΦ ⊂ R , which is called the attractor of Φ, such that F = j=1 φj(F ). Furthermore, FΦ is called a self-similar set if Φ consists of similitudes. It is well known that the standard middle-third Cantor set C is the attractor of the iterated function system (IFS) fφ0; φ1g where 1 1 2 (1.1) φ (x) = x; φ (x) = x + : 0 3 1 3 3 A natural question is: Is it possible to express C as the attractor of another IFS? Surprisingly, the general question whether the attractor of an IFS can be expressed as the attractor of another IFS, which seems a rather fundamental question in fractal geometry, has 1991 Mathematics Subject Classification.
  • Baire Sets, Borel Sets and Some Typical Semi-Continuous Functions

    Baire Sets, Borel Sets and Some Typical Semi-Continuous Functions

    BAIRE SETS, BOREL SETS AND SOME TYPICAL SEMI-CONTINUOUS FUNCTIONS KEIO NAGAMI Recently Hing Tong [3], M. Katetov [4] and C. H. Dowker [2] have es- tablished two sorts of insertion theorems for semi-continuous functions defined on normal and countably paracompact normal spaces. The purpose of this paper is to give the insertion theorem for some typical semi-continuous functions defined on a T-space. The relations between Baire sets and Borel sets in some topological spaces are also studied. Throughout this paper, unless in a special context we explicitly say other- wise, any function defined on a space is real-valued and a sequence of functions means a countable sequence of them. A family of functions defined on a space R is called complete if the limit of any sequence of functions in it is also con- tained in it. The minimal complete family of functions defined on R which con- tains all continuous functions is called the family of Baire functions and its ele- ment a Baire function. A subset of R is called a Baire set if its characteristic function is a Baire function. A subset of R is called respectively elementary- open or elementary-closed if it is written as {x fix) > 0} or {x I fix) ^ 0} by a suitable continuous function fix) defined on R. The minimal completely additive class of sets which contains all open sets in R is called the family of Borel sets and its element a Borel set. It is well known that the family of Borel sets contains the family of Baire sets and that the family of Baire sets coincides with the minimal completely additive class of sets which contains all the elementary open (closed) sets.
  • On Borel Measures and Baire's Class 3

    On Borel Measures and Baire's Class 3

    proceedings of the american mathematical society Volume 39, Number 2, July 1973 ON BOREL MEASURES AND BAIRE'S CLASS 3 R. DANIEL MAULDIN Abstract. Let S be a complete and separable metric space and fi a er-finite, complete Borel measure on S. Let <I>be the family of all real-valued functions, continuous /¿-a.e. Let Bx(^) be the func- tions of Baire's class a generated by O. It is shown that if ¡x is not a purely atomic measure whose set of atoms form a dispersed subset of S, then Bt(®)^Bai(<b), where a>i denotes the first uncountable ordinal. If O is a family of real-valued functions defined on a set S, then, B(<&), the Baire system of functions generated by O is the smallest subfamily of Rx which contains O and which is closed under the process of taking pointwise limits of sequences. The family 5(0) can be generated from <I> as follows: Let £„(<!>)=O and for each ordinal a, let -Ba(0) be the family of all pointwise limits of sequences taken from IJy<a ^O^O- Thus, B^ffb) is the Baire system of functions generated by <E>,where u>1 is the first uncountable ordinal. The Baire order of a family O is the first ordinal a such that BX(®)=BX+1(4>). Kuratowski has proved that if S is a metric space and O is the family of all real-valued functions on S which are continuous except for a first category set, then the order of í> is 1 and 2?i(<J>)is the family of all functions which have the Baire property in the wide sense [1, p.
  • Georg Cantor English Version

    Georg Cantor English Version

    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
  • Fractal Geometry and Applications in Forest Science

    Fractal Geometry and Applications in Forest Science

    ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals ....................................................................................
  • Fractals Lindenmayer Systems

    Fractals Lindenmayer Systems

    FRACTALS LINDENMAYER SYSTEMS November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea Behind Hidden Markov Models First letter: Maybe „a“, maybe „q“ Second letter: Maybe „r“ or „v“ or „u“ Take the most probable combination as a guess! Hidden Markov Models Sometimes, you don‘t see the states, but only a mapping of the states. A main task is then to derive, from the visible mapped sequence of states, the actual underlying sequence of „hidden“ states. HMM: A Fundamental Question What you see are the observables. But what are the actual states behind the observables? What is the most probable sequence of states leading to a given sequence of observations? The Viterbi-Algorithm We are looking for indices M1,M2,...MT, such that P(qM1,...qMT) = Pmax,T is maximal. 1. Initialization ()ib 1 i i k1 1(i ) 0 2. Recursion (1 t T-1) t1(j ) max( t ( i ) a i j ) b j k i t1 t1(j ) i : t ( i ) a i j max. 3. Termination Pimax,TT max( ( )) qmax,T q i: T ( i ) max. 4. Backtracking MMt t11() t Efficiency of the Viterbi Algorithm • The brute force approach takes O(TNT) steps. This is even for N = 2 and T = 100 difficult to do. • The Viterbi – algorithm in contrast takes only O(TN2) which is easy to do with todays computational means. Applications of HMM • Analysis of handwriting. • Speech analysis. • Construction of models for prediction.
  • Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets

    Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets

    Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set.
  • Set Theory: Cantor

    Set Theory: Cantor

    Notes prepared by Stanley Burris March 13, 2001 Set Theory: Cantor As we have seen, the naive use of classes, in particular the connection be- tween concept and extension, led to contradiction. Frege mistakenly thought he had repaired the damage in an appendix to Vol. II. Whitehead & Russell limited the possible collection of formulas one could use by typing. Another, more popular solution would be introduced by Zermelo. But ¯rst let us say a few words about the achievements of Cantor. Georg Cantor (1845{1918) 1872 - On generalizing a theorem from the theory of trigonometric series. 1874 - On a property of the concept of all real algebraic numbers. 1879{1884 - On in¯nite linear manifolds of points. (6 papers) 1890 - On an elementary problem in the study of manifolds. 1895/1897 - Contributions to the foundation to the study of trans¯nite sets. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather be- cause he initiated the study of in¯nite sets and numbers which have provided such fascinating material, and di±culties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one- to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and ¯ve papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after ¯nishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis.
  • Sets on Which Measurable Functions Are Determined by Their Range

    Sets on Which Measurable Functions Are Determined by Their Range

    Sets on which measurable functions are determined by their range Maxim R. Burke∗ Krzysztof Ciesielski† Department of Mathematics and Computer Science, Department of Mathematics, University of Prince Edward Island, West Virginia University, Charlottetown, P.E.I., Canada C1A4P3 Morgantown, WV 26506-6310, USA [email protected] [email protected] Version of 96/01/14 Abstract We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range. 1 Introduction In [BD, theorem 8.5], it is shown that, under the Continuum Hypothesis (CH), in any separable Baire topological space X there is a set M such that for any two continuous real-valued functions f and g on X,iff and g are not constant on any nonvoid open set then f[M] ⊆ g[M] implies f = g. In particular, if f[M]=g[M] then f = g. Sets havingthis last property with respect to entire (analytic) functions in the complex plane were studied in [DPR] where they were called sets of range uniqueness (SRU’s).We study the properties of such sets in measurable spaces with negligibles. (See below for the definition.) We prove a generalization of the aforementioned result [BD, theorem 8.5] to such spaces (Theorem 4.3) and answer Question 1 from [BD] by showingthat CH cannot be omitted from the hypothesis of their theorem (Example 5.17). We also study the descriptive nature of SRU’s for the nowhere constant continuous functions on Baire Tychonoff topological spaces. When X = R, the result of [BD, theorem 8.5] states that, under CH, there is a set M ⊆ R such that for any two nowhere constant continuous functions f,g: R → R,iff[M] ⊆ g[M] then f = g.Itis shown in [BD, theorem 8.1] that there is (in ZFC) a set M ⊆ R such that for any continuous functions f,g: R → R,iff has countable level sets and g[M] ⊆ f[M] then g is constant on the connected components of {x ∈ R: f(x) = g(x)}.
  • Generators, Recursion, and Fractals

    Generators, Recursion, and Fractals

    Generators, Recursion, and Fractals 1 Generators computing a list of Fibonacci numbers defining a generator with yield putting yield in the function fib 2 Recursive Functions computing factorials recursively computing factorials iteratively 3 Recursive Images some examples recursive definition of the Cantor set recursive drawing algorithm MCS 260 Lecture 41 Introduction to Computer Science Jan Verschelde, 22 April 2016 Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 1 / 36 Generators, Recursion, and Fractals 1 Generators computing a list of Fibonacci numbers defining a generator with yield putting yield in the function fib 2 Recursive Functions computing factorials recursively computing factorials iteratively 3 Recursive Images some examples recursive definition of the Cantor set recursive drawing algorithm Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 2 / 36 the Fibonacci numbers The Fibonacci numbers are the sequence 0, 1, 1, 2, 3, 5, 8,... where the next number in the sequence is the sum of the previous two numbers in the sequence. Suppose we have a function: def fib(k): """ Computes the k-th Fibonacci number. """ and we want to use it to compute the first 10 Fibonacci numbers. Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 3 / 36 the function fib def fib(k): """ Computes the k-th Fibonacci number. """ ifk==0: return 0 elif k == 1: return 1 else: (prevnum, nextnum) = (0, 1) for i in range(1, k): (prevnum, nextnum) = (nextnum, \ prevnum + nextnum) return nextnum Intro to Computer Science (MCS 260) generators and recursion L-41 22 April 2016 4 / 36 themainprogram def main(): """ Prompts the user for a number n and prints the first n Fibonacci numbers.