Introduction to Real Analysis and Fourier Analysis Mijia Lai updated on March 8,2020
Introduction to Real Analysis and Fourier Analysis Mijia Lai updated on March 8, 2020
Contents 55 1.2 Cardinality 1.3 Topology of the Euclidean space 5c and Digory the 901 1.5.1 Hausdorff distance and Gromoy-Hausdorff distance 1.5.2 Invariant of domain............... 2 Lebesgue measure 2.1 Exterior measure 。。 asurable ets 92 2.5 Sets of positive measure 3.1 urable 41 3.2 Simple functions 3.3 Littlewood'sThree principles,.,,,,,,, 30 4 Lebesgue's integration theory 4.】Integration nterchanging limits with integrals 4.4 Fubini's Theore 5 Differentiatic 5.2 Fundamental theorem of Calculus I 5.2.1 A detour:Bounded variation funetions......................50 6 Function spaces 6.1 6.11 6.12 A detour:Conveity ad Jense'inquaty 6.1.3 Completeness:Banach space... 62 3
Contents 1 Preliminary 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Topology of the Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Metric space and Baire Category theorem . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Continuous functions and Distance in metric space . . . . . . . . . . . . . . . . . . . 11 1.5.1 Hausdorff distance and Gromov-Hausdorff distance . . . . . . . . . . . . . . . 13 1.5.2 Invariant of domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Lebesgue measure 17 2.1 Exterior measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Borel sets and Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Linear transformation of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Sets of positive measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Measurable functions 27 3.1 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Littlewood’s Three principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Lebesgue’s integration theory 33 4.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Interchanging limits with integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Lebesgue v.s. Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Differentiation 45 5.1 Monotone functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Fundamental theorem of Calculus I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 A detour: Bounded variation functions . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Fundamental theorem of Calculus II . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Lebesgue Differentiation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6 Function spaces 59 6.1 L P spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1.1 Normed vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1.2 A detour: Convexity and Jensen’s inequality . . . . . . . . . . . . . . . . . . 61 6.1.3 Completeness: Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.1.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3
4 CONTENTS 6.2H 62.10pac9 6.2.2 6 6.2.3 Linear functional.Duality............ 5 7 Fourier Series 7.1 ntroduction· 7202 twise convergence 7.2.2 Abel summation 072 7.3 L2 convergence 7.4 Applications 72 8 Fourier Transforms 8.1.1 Fourier transform on S(R) 777 8.1.2 Inversion formula 82 rm on 8.3.1 Heat equation on R 9 Selected topics 9. 11 finite 912 Euler product formula 9.2 Falconer conjecture..... 88858888 92.4 Fourier transform to measure 93 of large numbers and Centra limit theorem 。 9.3.3 Central limit theorem
4 CONTENTS 6.2 Hilbert space: L 2 spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.1 Inner product and Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.2 Orthogonality, Orthonormal basis, Fourier series . . . . . . . . . . . . . . . . 64 6.2.3 Linear functional, Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7 Fourier Series 69 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.2.1 Ces`aro summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2.2 Abel summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.3 L 2 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4.1 Isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4.2 Weyl’s equidistribution theorem . . . . . . . . . . . . . . . . . . . . . . . . . 74 8 Fourier Transforms 77 8.1 Fourier transform on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.1 Fourier transform on S(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.1.2 Inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.1.3 The Plancherel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.2 Fourier transform on R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3.1 Heat equation on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3.2 Harmonic functions on upper half plane . . . . . . . . . . . . . . . . . . . . . 82 8.3.3 Wave equation in R n × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9 Selected topics 83 9.1 Dirichlet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.1.1 Fourier analysis on finite group . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9.1.2 Euler product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.2 Falconer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9.2.1 Hausdorff measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 9.2.2 Falconer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.2.3 Abstract Borel measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.2.4 Fourier transform to measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.3 Law of large numbers and Central limit theorem . . . . . . . . . . . . . . . . . . . . 90 9.3.1 A crash course in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9.3.2 Law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 9.3.3 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter 1 Preliminary 行路难！行路难！多歧路，今安在？ 长风硫浪会有时直辈否膏露潜 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis.It can into two parts.T hrst part the tegatioi number of discontimous pointsis countable.Therefore the integr almost continuous functions.Even though the great triumph was achieved by the Riemannian stha Indeed continuous functions are convergent to f then 1.f may not be Riemannian integrable; 2.even f is Riemannian integrable, 。in(h=厂feld may not hold. We give a counter-example for item 1 in the above.We can enumerate all rational numbers in [0,1]as (a,q2.........define 1a)-{bm0…9 It follows that f converges to the Dirichlet function D(),which is not Riemannian integrable. 5
Chapter 1 Preliminary 1¥Jú1¥Júı‹¥ß8S3º ºªL¨kûßÜ! ~LÙ°" ))ox51¥J6 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis. It can be roughly divided into two parts. The main subject in the first part is the Lebesgue’s integration theory. We have learned in Calculus that a function is Riemannian integrable if and only if the number of discontinuous points is countable. Therefore the Riemannian integral mainly works with almost continuous functions. Even though the great triumph was achieved by the Riemannian integral, it still has a major defect: not working well with limit. Indeed, continuous functions are not closed under taking limit, i.e., the limit of sequence of continuous functions is not necessarily continuous. Moreover, let fn be a sequence of Riemannian integrable functions on [0, 1], which is convergent to f then 1. f may not be Riemannian integrable; 2. even f is Riemannian integrable, limn→∞ Z 1 0 fn(x)dx = Z 1 0 f(x)dx may not hold. We give a counter-example for item 1 in the above. We can enumerate all rational numbers in [0, 1] as {q1, q2, · · · , ...}, define fn(x) = 1, x = q1, q2, · · · , qn; 0, else. It follows that fn converges to the Dirichlet function D(x), which is not Riemannian integrable. 5
6 CHapTER L PRELIMINARY in vals (cubes). underlining Euclidean geometry,on the other hand,put strong restrictions onto the local behavior of emanian integral represents theare e curve,th strips.Le horizontal strip may spread everywhere,however,it turns out to be a sweet surprise.As the local behavior of th in consideratio e care This viewpoint dramatically enlarges the range of integrable functions.The corresponding inte gral theory now boils down to the definition of the measure,and the rest follows alm nost naturally Another great advantage of reat convenience in subiect such as probability theory. -wise,in this course we shall provide the following generalization: length,.area,volume,…→measure continuous functions =measurable functions Riemannian integral=Lebesgue integral n the following,we sketch some important historical moments of the development for the real
6 CHAPTER 1. PRELIMINARY The basic idea of Riemannian integral is to divide the domain of definition into small intervals (cubes for higher dimensions). These neighboring intervals (cubes), on the one hand, rely on the underlining Euclidean geometry, on the other hand, put strong restrictions onto the local behavior of integrable functions. (cannot oscillate too much, thus leading to the continuity to some extent) The geometric meaning of the Riemannian integral represents the area under the curve, thus Riemann’s way of integration, roughly speaking, is to approximate the area by dividing the region into vertical strips. Lebesgue’s viewpoint is to view the region by horizontal strips. At a first glance, each horizontal strip may spread everywhere, however, it turns out to be a sweet surprise. As the local behavior of the function in consideration is not so critical, and what really matters now is the set of the form {f ≥ c}, which motivates the careful definition of its measure (strictly speaking, in this book by measure we mean Lebesgue measure). This viewpoint dramatically enlarges the range of integrable functions. The corresponding integral theory now boils down to the definition of the measure, and the rest follows almost naturally. Another great advantage of Lebesgue’s integral theory is that it is not restricted only to the integration on Euclidean space. It can equally be transplanted to any abstract measure space, yielding great convenience in subject such as probability theory. We shall see the above counter-example holds true in the sense of Lebesgue’ integration. Namely, the Dirichlet function is Lebesgue integrable and our hope that limn→∞ R [0,1] fn(x)dx = R [0,1] D(x)dx becomes true. Vocabulary-wise, in this course we shall provide the following generalization: length, area, volume, ... =⇒ measure continuous functions =⇒ measurable functions Riemannian integral =⇒ Lebesgue integral In the following, we sketch some important historical moments of the development for the real analysis
Some Historical developments of real analysis Weierstrass's nowhere 1872 differentiable function Introduction of 1881 functions by Jordan and 1883 Cantor set later connection with rectifiability 1890 Space filling curve by Peano 1898 Borel's measurable sets 1902 bebesgue's theory of Consfruction 1905 measure and integration non-measurable sefs by Vitali 2
2 Some Historical developments of real analysis Weierstrass’s nowhere differentiable function 1872 Introduction of BV functions by Jordan and later connection with rectifiability Cantor set Space filling curve by Peano Construction of non-measurable sets by Vitali Borel’s measurable sets Lebesgue’s theory of measure and integration 1881 1883 1890 1898 1902 1905
8 CHAPTER 1.PRELIMINARY The othtThererexpectd oction of Fourer 1.2 Cardinality In following sections,we establishsome foundations on the set theory and the topology and geometry the E t this miliar with infinite elements?This requires the concept of the cardinality of a set For two sets with finite number of elements,it is clear which set contains more elements.For two sets with in me e if V2 B.th f()=2.A mapf:ABis called a bijection if f is both injective and surjective.Clearly,a map f:AB has a well-defined inverse,if and only if f is a bijection B are called to tion The cardinal number of natural numbers N is denoted by o.(Countable) Erample 1.Each infinite set contains a countable subset Erample 2.Countable uion of countable sets is countable. Proof.Array this union as an infinite square,and enumerate in a zigzag way. 0 Erample 3.All rational numbers Q is countable. Erample 4.Finite cartesian product of countable sets is countable Proof.Visualize this union as an infinitek dimensional cube,and enumerate in a zigzag way. ◇ Erample 5.The set of all real numbers R is not countable. (0.1l is not countable.We accept each real n om position.That we write0.25as0.249999999,1as0.99999. ete Now suppos (0,1]is countabl then we have an en meration for all mbers in(0,，小say that y is in The cardinality ofR is called The decimal representation shows that countable product of finite sets has cardinal number Erample 6.R,(0,1],(0,1],R all have same cardinal number Theorem 1.1.There docs not erist marimal cardinal number Proof.Given any set A.consider its power set 24.namely the set of all subsets of A.We ca show they have different cardinality.Otherwise,there exists a bijectionf:A24,where f(a) corresponds to a subset of A.Define a subset of A as follows: B={xz度f(z
8 CHAPTER 1. PRELIMINARY The second part begins with the rudiment of the function spaces, followed by an introduction to Fourier analysis. We study both Fourier series and Fourier transform together with their applications. The connection with real analysis is intimacy. There are also many unexpected connections of Fourier analysis to wide-ranging mathematical topics such as Number theory, Discrete geometry, Probability theory. We convey to the reader only a small portion of this fascinating subject. 1.2 Cardinality In following sections, we establish some foundations on the set theory and the topology and geometry of the Euclidean space. We assume the reader is familiar with basic notions of sets, operations between sets, etc. In this section, we address the following question: how to compare two sets with infinite elements? This requires the concept of the cardinality of a set. For two sets with finite number of elements, it is clear which set contains more elements. For two sets with infinite elements, which contains ’more’ elements relies on the mappings between them. A map f : A → B is an assignment to each element of A a unique element in B. f is called injective, if f(x) 6= f(y), for x 6= y. f is called surjective if ∀z ∈ B, there exists x ∈ A such that f(x) = z. A map f : A → B is called a bijection if f is both injective and surjective. Clearly, a map f : A → B has a well-defined inverse, if and only if f is a bijection. A and B are called to have same cardinality if there exists a bijection f : A → B, denoted by A ∼ B. Sometimes, we shall refer to the cardinal number of a set A, denoted by A ¯¯. The cardinal number of natural numbers N is denoted by ℵ0. (Countable) Example 1. Each infinite set contains a countable subset. Example 2. Countable union of countable sets is countable. Proof. Array this union as an infinite square, and enumerate in a zigzag way. Example 3. All rational numbers Q is countable. Example 4. Finite cartesian product of countable sets is countable. Proof. Visualize this union as an infinite k dimensional cube, and enumerate in a zigzag way. Example 5. The set of all real numbers R is not countable. Proof. We prove (0, 1] is not countable. We accept each real number in (0, 1] has a decimal representation, which is unique if we don’t allow the appearance of all zeros after some position. That is we write 0.25 as 0.249999999..., 1 as 0.99999...., etc. Now suppose (0, 1] is countable, then we have an enumeration for all numbers in (0, 1], say 0.a11a12a13...., 0.a21a22a23..., ... We can choose bii ∈ {0, 1, 2, ..., 9} \ aii, for each i. Let y = 0.b11b22b33..., a moment of thought shows that y is indeed not in the enumeration list. A contradiction. The cardinality of R is called ℵ1. The decimal representation shows that countable product of finite sets has cardinal number ℵ1. Example 6. R, (0, 1], [0, 1], R n all have same cardinal number ℵ1. Theorem 1.1. There does not exist maximal cardinal number. Proof. Given any set A, consider its power set 2A, namely the set of all subsets of A. We can show they have different cardinality. Otherwise, there exists a bijection f : A → 2 A, where f(a) corresponds to a subset of A. Define a subset of A as follows: B = {x|x /∈ f(x)}
1.3.TOPOLOGY OF THE EUCLIDEAN SPACE Now an is B=f(r)for mex∈A? This of the barber paradox which was raised by Bertrand Russellas There is no set whose cardinality is strictly between that of the integers and the real numbers. ipedia. 1.3 Topology of the Euclidean space We use Ra for n-dimensional Euclidean space.For =(1..andy=(.),the inner product is defined as 工·y=r1边+r22+··+xmn Norm is defined as 回=√+…+ Open ball centered at r of radius r is denoted by B(.r).i.e., B(z.r)=uy- nice property of being a compact setis that any open cover hasa finite subcover. Theneml3但eineBare.AcRrwaompdtfamdomtfeenyopeacorgofAcontains We also recall the theorem of nested closed sets. Theorem 1.4.Let AAA..be a sequence of nested non-emply closed sets.Then ∩e1A≠0. nle 8 (Cantor set).Let co =o 11 the unit closed int moval of the middle ne third open intervals of each connected components of Cn-1.For example,C2=0,1]. c=∩c. is the Cantor set
1.3. TOPOLOGY OF THE EUCLIDEAN SPACE 9 Now an amusing question confronts us: is B = f(x) for some x ∈ A? This proof is reminiscent of the barber paradox, which was raised by Bertrand Russell as follows: a barber in a town claims to be the ”one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave himself? Remark 1.2 (Continuum hypothesis). Cantor in 1878 raised the following hypothesis concerning the size of infinite sets: There is no set whose cardinality is strictly between that of the integers and the real numbers. Establishing its truth or falsehood is the first of Hilbert’s 23 problems presented in 1900. The reader is referred to https://en.wikipedia.org/wiki/Continuum hypothesis for a thorough introduction. 1.3 Topology of the Euclidean space We use R n for n-dimensional Euclidean space. For x = (x1, · · · , xn and y = (y1, · · · , yn), the inner product is defined as x · y = x1y1 + x2y2 + · · · + xnyn. Norm is defined as |x| = q x 2 1 + · · · + x 2 n . Open ball centered at x of radius r is denoted by B(x, r), i.e., B(x, r) = {y||y − x| 0 such that B(x, r) ⊂ A. A is called an open set, if every point of A is an interior point. x is called an accumulation point of A, if (B(x, r) \ {x}) ∩ A 6= ∅, for all r > 0. The union of A with its accumulation points is called the closure of A, denoted by A¯. A set A is called closed, if A¯ is an open set. A family of open sets {Oα}α∈Λ is called an open cover of A if A ⊂ S α Oα. A is bounded if there exists R > 0, such that A ⊂ B(0, R). A set is called compact if it is both bounded and closed. A nice property of being a compact set is that any open cover has a finite subcover. Theorem 1.3 (Heine-Borel). A ⊂ R n is a compact set if and only if every open cover of A contains a finite subcover. We also recall the theorem of nested closed sets. Theorem 1.4. Let A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · be a sequence of nested non-empty closed sets. Then T∞ n=1 An 6= ∅. B ⊂ A is called dense in A, if B¯ = A. A is called nowhere dense if there exists no interior point of A¯. Example 7. Take r /∈ Q, denotes the fractional part of x. Then {}n=1,2,··· is dense in [0, 1]. Example 8 (Cantor set). Let C0 = [0, 1] the unit closed interval. C1 = [0, 1 3 ] ∪ [ 2 3 , 1], the removal of the middle 1 3 open interval from C0. Cn is obtained inductively by removing the middle one third open intervals of each connected components of Cn−1. For example, C2 = [0, 1 9 ]∪[ 2 9 , 1 3 ]∪[ 2 3 , 7 9 ]∪[ 8 9 , 1]. C := \∞ n=0 Cn is the Cantor set
CHAPTER 1.PRELIMINARY 国 翻铜 Figure 1.1:Cantor set The following proposition lists several properties of the Cantor set. Proposition 1.5.The Cantor set C defined as above is non-empty and satisfies the following properttes. ·C is closed. .C does not contain any interior point,hence it is nowhere dense. .C is uncountable,and its cardinal number is closed. Suppose C is an interior point,then there exists>0,such that (+)CC.Taking N large enoug such that her with clo cmC.folo here dens Using the decimal representation of base 3 for all real numbers in [0,1],i.e,=,where inwedon'tall the tha di= the has the cardinal number 1.4 Metric space and Baire Category theorem Given a set X,a map d:xR satisfying 1.Symmetry d(,)=d(y,); 2.Positivity d()0 and =holds if and only if=y: 3.Triangle inequality d(z,)+d(,z)≥d(工，z is called a metric on X.(X,d)is then called a metric space. tion of convergence.limn=z if and only if limd(n,) Ve >0,there exists N,such that d(rn,rm)se,Vn,m N. A metric space is called complete if any Cauchy sequence is convergent in the space.The concepts of open balls,open sets,closed sets,interior points,closure,etc,all generalize to the metric space. Theorem 1.6(Baire Category Theorem).A non-emply complete metric space is not a counable DSimilaly D)s nonempty open set,can chocse suc that B
10 CHAPTER 1. PRELIMINARY Figure 1.1: Cantor set The following proposition lists several properties of the Cantor set. Proposition 1.5. The Cantor set C defined as above is non-empty and satisfies the following properties: • C is closed. • C does not contain any interior point, hence it is nowhere dense. • C is uncountable, and its cardinal number is ℵ1. Proof. C is not empty. A moment of thought shows that the end points of those middle third intervals all remain in C. Since each Cn is closed, the intersection of countable closed sets is still closed. Suppose x ∈ C is an interior point, then there exists δ > 0, such that (x − δ, x + δ) ⊂ C. Taking N large enough such that 1 3N 0, there exists N, such that d(xn, xm) ≤ , ∀n, m > N. A metric space is called complete if any Cauchy sequence is convergent in the space. The concepts of open balls, open sets, closed sets, interior points, closure, etc, all generalize to the metric space. Theorem 1.6 (Baire Category Theorem). A non-empty complete metric space is not a countable union of nowhere dense sets. Proof. Suppose not. Then assume X = S∞ n=1 Dn, where each Dn is a nowhere dense set. Clearly X \ D1 is not empty, therefore there exists an interior point x1 and 1 > 0 such that B(x1, 1) ⊂ X \ D1. Similarly D2 c ∩ B(x1, ) is a nonempty open set, we can choose x2, 2 such that B(x2, 2) ⊂
1.5.CONTINUOUS FUNCTIONS AND DISTANCE IN METRIC SPACE 11 ce of nested balls B( )C B( 1).mor equences and it cor rerges to. Using the Baire category theorem,we get another proof that 0,I is uncountable. Gs set,countable union of closed sets is called an 0ion7.Phere dow not erist a mction加.广：R→hich t continous ony t7 We need a lemma first Lemma 1.8.The points of continuity of f is a Gs set. Proof.Recall that f is continuous at z if and only if the oscillation wf()=0.Therefore the set of points of continuity of f is ∩wr回)0 such that If(y)-fr川≤6，y∈B(x,6)nE. f is called continuous on E if f is continuous at every point of E.This definition does not require E is open. Theorem 1.9.Suppose f:FR be a continuous function defined on a compact set F,then f is uniform continuous and attains its marimum and minimum
1.5. CONTINUOUS FUNCTIONS AND DISTANCE IN METRIC SPACE 11 D2 c ∩ B(x1, ). Inductively, we get a sequence of nested balls B(xn, n) ⊂ B(xn−1, n−1), moreover we can easily arrange that limn→∞ n = 0. Thus {xn} is a Cauchy sequences and it converges to, say x. Since X = S∞ n=1 Dn, thus x ∈ Dk for some k. However due to the construction x ∈ B(xk, k), which contradicts to that B(xk, k) ∩ Dk = ∅. Using the Baire category theorem, we get another proof that [0, 1] is uncountable. Countable intersection of open sets is called a Gδ set, countable union of closed sets is called an Fσ set. We give a more interesting application of Baire’s category theorem. Proposition 1.7. There does not exist a function f : R → R which is continuous only at all rational numbers. We need a lemma first. Lemma 1.8. The points of continuity of f is a Gδ set. Proof. Recall that f is continuous at x if and only if the oscillation ωf (x) = 0. Therefore the set of points of continuity of f is \∞ n=1 {x|ωf (x) 0, there exists δ > 0 such that |f(y) − f(x)| ≤ , ∀y ∈ B(x, δ) ∩ E. f is called continuous on E if f is continuous at every point of E. This definition does not require E is open. Theorem 1.9. Suppose f : F → R be a continuous function defined on a compact set F, then f is uniform continuous and attains its maximum and minimum