Coarse functions are functions whose graphs appear to be continuous at a distance,   but in fact may not be continuous. In this paper we explore examples and properties of coarse functions.   We then generalize several basic theorems of continuous functions which apply to coarse functions

Duke, Jared

Lee, Chul-Woo

English

Rose-Hulman Institute of Technology: Rose-Hulman Scholar

Rose-Hulman Undergraduate Mathematics Journal Volume 8 Issue 2 Article 4 Course Function Value Theorems Jared Duke Brigham Young University, glorindel@hotmail.com Chul-Woo Lee Brigham Young University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Duke, Jared and Lee, Chul-Woo (2007) "Course Function Value Theorems," Rose-Hulman Undergraduate Mathematics Journal: Vol. 8 : Iss. 2 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol8/iss2/4 COARSE FUNCTION VALUE THEOREMSCHUL-WOO LEE AND JARED DUKEAbstract. Coarse functions are functions whose graphs appear to be contin-uous at a distance, but in fact may not be continuous. In this paper we exploreexamples and properties of coarse functions. We then generalize several basictheorems of continuous functions which apply to coarse functions.1. IntroductionIn this paper we will explore several properties of coarse functions. In particular,we will establish the conditions necessary to generalize several basic theorems ofcalculus for use with such functions. Unlike continuous functions, where attentionis placed on the small scale structure defined by a given metric, coarse functionsfocus instead on the large scale structure of a function. This large scale focus isthe defining aspect of coarse geometry, where the coarse equivalence of two metricspaces is established by their similarity when viewed from a great distance [1].In the stock market, prices are updated at discrete intervals, namely every sec-ond. Such sampling gives us a discrete set of data rather than some continuous func-tions. However, because we are more concerned with the long-term price trends ofsuch stocks, whether that be hourly, daily, weekly or monthly, we need not concernourselves with the small gaps in our data set. Though there exists discontinuitybetween the stock prices on our second to second intervals, we can ”take a stepback” and find coarse continuity as we look at the prices on larger and larger inter-vals. This gives us the information we need, as we can use the principles of coarsegeometry to analyze and interpret such data. The graphs in Figure 1 represent DowJones Industrial Average closings taken from two time periods, one short term andthe other long term. The short term data on the left appears rough and unrelated,while the long term data on the right appears smooth and can be approximated bya continuous function over R2.Figure 1. Stock Prices12 CHUL-WOO LEE AND JARED DUKEIn engineering, these are the cases where functions are defined over a discretedomain but viewed in large scale. The material properties of a specific model’s mi-crostructure, for example, are defined only at certain points on the given structure,a ”grain space” [2]. The application of coarse geometric principles to this discreteset of data allows us to make useful observations and conclusions about the model’sbehavior and properties, as the data can be treated more like a continuous function.The appearance of perfect, continuous functions in real world situations is rare.More common is the appearance of discrete data sets. This data contains much use-ful information. However, making sense of that information proves a more difficulttask. Take, for example, the terrain elevation samplings shown in Figure 2. Thevertices of the piecewise linear surface on the left represent a small scale samplingfrom the terrain data. Note the relative jaggedness of the graph. The image on theright, however, is a much larger sampling from the same terrain data. Though stilldiscrete, the graph appears smoother and significantly less jagged.Figure 2. Terrain ModelThe purpose of this paper is to create a broader mathematical basis for inter-preting such discrete sets of data. The figures on the following pages are indicativeof the types of data sets for which such analysis would prove useful.Coarse functions and geometry, in the context of geometric group theory, havebeen extensively studied by John Roe of Penn State University. Please refer to [3]for more information on the subject. This paper relies on several fundamental ideasbehind coarse-geometry, primarily the characterization of functions as large-scaleLipschitz, bornologous and weak-bornologous as found in [3]. By establishing theequivalence of these conditions under certain conditions, we created a framework foradapting fundamental theorems for continuous functions, as found in most calculustextbooks, such as Stewart’s Calculus [4]. We then prove these adapted theoremsfor coarse functions.2. PreliminariesThe following definition, as found in [3], provides the basis for our work withcoarse functions.Definition 2.1. Let X and Y be metric spaces, and let f : X → Y be a map.Then for x, x′ ∈ X we define the following:(1) The map is large-scale Lipschitz if there exist positive constants c and Asuch thatd(f(x), f(x′)) ≤ cd(x, x′) +ACOARSE FUNCTION VALUE THEOREMS 3Note that if A = 0, we have that the function satisfies the Lipschitz condi-tion, which in turn implies that f is continuous. However, since our focuswill be on discrete functions, then in general we will find that A 6= 0. Anexample of such a map can be found in Figure 3. Let f be defined asfollows:f(x) ={√x2 − 3 for |x| > 2− |x|+ 2 for |x| ≤ 2Then we find that f satisfies the large-scale Lipschitz condition with c = 2and A = 2.Figure 3. Large-Scale Lipschitz Mapping(2) The map f is bornologous if for every R > 0 there exists an S > 0 suchthatd(x, x′) < R⇒ d(f(x), f(x′)) < S.Consider the function f(x) = x2 on the set A = {1/n | n ∈ Z+}. Now, letR > 0 be given and let S = R. So for any x, x′ ∈ A ⊂ (0, 1), we have bythe definition of f(x) that d(f(x), f(x′)) < d(x, x′) < R = S. Clearly then,we have that f is a bornologous function on the discrete set A.(3) The map is weak bornologous if there exist R,S > 0 such thatd(x, x′) < R⇒ d(f(x), f(x′)) < SA weak bornologous mapping can be found in Figure 4. Let f be definedas follows:f(n) ={2 for x = ± 12n2 for n ∈ ZFor R = 1 and S = 3 we find that f satisfies the weak bornologous condi-tion. Note, however, that f is not bornologous.4 CHUL-WOO LEE AND JARED DUKEFigure 4. Weak Bornologous MappingIt will be shown that these conditions are in fact equivalent under certain restric-tions. Such equivalence allows us to expand several conclusions about continuousfunctions to those that are coarse and discrete. Before doing so, however, we providethe definitions of those terms we created as tools for expanding these conclusions.Definition 2.2. A δ-coarse path P from points a, b ∈ X with δ > 0 is an orderedset P = {a = x0, x1, . . . , xn−1, xn = b} such that d(xi−1, xi) < δ for 1 ≤ i ≤ n.Now, let a = (0, 0) and b = (4, 2) be points in R2. Clearly, there are many δ-coarsepaths between a and b. Figure 5 shows such a path with δ = 2.Figure 5. δ Coarse PathDefinition 2.3. A δ-γ coarse path P from a to b is a δ-coarse path from a to bgiven by P = {a = x0, x1, . . . , xn−1, xn = b} such that δ − 2γ ≤ d(xi−1, xi) < δ for1 ≤ i ≤ n. A δ-γ coarse path is dependent on the chosen values of both delta andgamma. For example, the two previous paths shown in Figure 5 are not δ-γ coarsepaths for δ = 2 and γ = .5. However, if we reduce our constraints and let δ = 1.5and γ = .3, then the paths satisfy the necessary conditions for a δ-γ coarse path.Figure 6 illustrates a δ-γ coarse path with δ = 1 and γ = .3.Definition 2.4. Let f : X → R be defined on a metric space X. We define a coarsederivative function f ′δ(x) on X as follows. Suppose P = {a = x0, x1, . . . , xi =COARSE FUNCTION VALUE THEOREMS 5Figure 6. δ-γ Coarse Pathx, xi+1, . . . , xn−1, xn = b} is a δ-coarse path from a to b in X. Thenf ′δ(xi) =f(xi+1)− f(xi)δNote that this definition better mimics the standard definition of a derivative if Pis a δ-γ course path for δ very small and γ << δ.Definition 2.5. A set X in a metric space is bounded if there exists an R < ∞such that d(x, x′) ≤ R for all x, x′ ∈ S.3. The Large Scale Lipschitz Function Theorem for CoarseFunctionsWe begin by establishing the restrictions under which the weak bornologous con-dition implies large-scale Lipschitz behavior for discrete functions. In generalizingthis relationship for use in a discrete environment, we gain the ability to applyadditional properties of coarse geometry already established in [3] .Theorem 3.1. Let X and Y be metric spaces and f : X → Y a function such that:(1) f is weak bornologous where R and S are the constants so that d(x, x′) < Rimplies that d(f(x), f(x′)) < S.(2) There exists an  > 0 such that if x, y ∈ X, d(x, y) > R, and n is an theinteger so thatd(x, y)n< R − 2 and d(x, y)n− 1 ≥ R − 2, then there is aδ-course path P = {x = x0, x1, . . . , xn = y} with δ = R.Then f is large-scale lipschitz.Proof. Let x, y ∈ X. Let R,S and  be given as in (1) and (2). Choose n so thatd(x, y)n< R− 2 and d(x, y)n− 1 ≥ R− 2.6 CHUL-WOO LEE AND JARED DUKEThus we have that n ≤ d(x,y)(R−2) + 1. Now let x0 = x, x1, . . . , xn−1, xn = y be givenfrom (2). Thend(f(x), f(y)) ≤n−1∑i=0d(f(xi), f(xi+1))< nS≤(d(x, y)R− 2 + 1)S=(SR− 2)d(x, y) + SThus d(f(x), f(y)) <(SR−2)d(x, y) + S. Therefore, f is large scale Lipschitz.Corollary 3.2. Let X and Y be metric spaces and let f : X → Y satisfy therestrictions set forth in Theorem 3.1. Then the following statements are equivalent:(1) f is large-scale Lipschitz(2) f is bornologous(3) f is weak bornologousProof. We begin by showing that (1) ⇒ (2). Suppose f is large-scale Lipschitz.Then there exist positive constants c and A such thatd(f(x), f(y))) ≤ cd(x, y) +Afor all x, y ∈ X. Now let R > 0 be given, and choose S = cR + A. Thus, ifd(x, y) < R we have thatd(f(x), f(y)) ≤ cd(x, y) +A< cR+A = SNow for (2)⇒ (3). Suppose f is bornologous. Thus for every R > 0 there existsan S > 0 such thatd(x, x′) < R⇒ d(f(x), f(x′)) < S.Clearly, there must then exist some R > 0 and S > 0 such that the same conditionholds. Thus we have that f is weak bornologous.Finally, (3)⇒ (1) is proved in Theorem 3.1.4. The Intermediate Value Theorem for Coarse FunctionsTheorem 4.1. Let X be a metric space and f : X → R be weak bornologous. LetP = {a = x0, x1, . . . , xn = b} be a δ-coarse path from a to b in X with δ = R.Then, if y is a value strictly between f(a) and f(b), there exists c ∈ P such thatd(f(c), y) < S.Proof. Without loss of generality, suppose that f(a) ≤ f(b) and let A = {xi ∈P |f(xi) ≤ y}. Note that A 6= ∅ because x0 = a ∈ A. Also A 6= P becausexn = b /∈ A. Now, since P is a set of ordered points, so too will A be a setCOARSE FUNCTION VALUE THEOREMS 7of ordered points. Let k be the maximum index of the xi’s in A. Then, clearly,f(xk) ≤ y < f(xk+1). Thusd(f(xk), y) ≤ d(f(xk), f(xk+1)) < Sd(f(xk), y) < STherefore, c = xk is the desired point. 5. The Boundedness Theorem for Coarse FunctionsThe following theorem is the coarse analog for the Extreme Value Theorem.Theorem 5.1. Let f : X → R be real valued function defined on a bounded set Psuch that f is large scale Lipschitz. Then the image of P under f is also bounded.Proof. Let P be a bounded subset of X. So for any x, y ∈ P , d(x, y) < R, whereR <∞. Now, since f is large-scale Lipschitz, there exists positive constants c andA such thatd(f(x), f(y)) ≤ cd(x, y) +A< cR+ASo by definition, the image of P is also bounded. 6. The Mean Value Theorem for Coarse FunctionsTheorem 6.1. Let f : X → R be defined such that f ′δ is large scale lipschitz withconstants c and A. Also, let P be a well-defined δ coarse path from x to y forx, y ∈ X. Then there exists some xi ∈ P such that, for β = 12 (cδ +A):∣∣∣∣f(y)− f(x)d(y, x) − f ′δ(xi)∣∣∣∣ ≤ βProof. Let x, y ∈ X, and let f and P be given as in the hypothesis. Now, assumeto the contrary that, for all xi ∈ P for which f ′δ exists, the following holds:∣∣∣∣f(y)− f(x)d(y, x) − f ′δ(xi)∣∣∣∣ > βSo we have, for all appropriate xi, that either(1) f ′δ(xi) >f(y)− f(x)d(y, x)+ βor(2) f ′δ(xi) <f(y)− f(x)d(y, x)− βNow, assume without loss of generality that equation (1) holds for x1. It followsthat the same equation must hold for all remaining xi ∈ P . Otherwise, there wouldexist some xi, xi−1 ∈ P such thatf ′δ(xi−1) >f(y)− f(x)nδ+ βandf ′δ(xi) <f(y)− f(x)nδ− βThis would imply thatf ′δ(xi−1)− f ′δ(xi) > 2β8 CHUL-WOO LEE AND JARED DUKEHowever, since f ′δ is large scale lipschitz, then for any xi, xi−1 ∈ P ,|f ′δ(xi)− f ′δ(xi−1)| ≤ cd(xi, xi−1) +A < cδ +A = 2βThis contradiction implies that equation (1) must hold for all xi ∈ P if it holds forx1. Now, note that d(y, x) ≤ nδ. It follows from equation (1) thatn−1∑i=0f ′δ(xi) > (n)(f(y)− f(x)d(y, x)+ β)(3)≥ (n)(f(y)− f(x)nδ+ β)=f(y)− f(x)δ+ nβBy our definition of f ′δ(xi), however, we know thatn−1∑i=0f ′δ(xi) =f(xn)− f(xn−1) + f(xn−1)− . . .− f(x1) + f(x1)− f(x0)δ=f(xn)− f(x0)δ=f(y)− f(x)δ(4)Substituting (4) into the left hand side of equation (3) gives us the following:f(y)− f(x)δ>f(y)− f(x)δ+ nβand therefore0 > nβ =(n2)(cδ +A)This is a contradiction, however, as n, c, δ and A are all nonnegative constants.Thus, it must be the case that there exists some xi ∈ P such that:∣∣∣∣f(y)− f(x)d(y, x) − f ′δ(xi)∣∣∣∣ ≤ βReferences[1] J. Roe. What is a Coarse Space?.[2] Adams, Kalidindi, and Fullwood. Microstructure Sensitive Design for Performance Opti-mization. BYU Academic Publishing, 2005.[3] J. Roe. Lectures on Coarse Geometry. American Mathematical Society, 2003.[4] J. Stewart. Calculus. Brooks Cole, 2002.

Course Function Value Theorems

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Course Function Value Theorems

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