Counterexamples in analysis

Description

These counterexamples deal mostly with the part of analysis known as "real variables." The 1st half of the book discusses the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, more. The 2nd half examines functions of 2 variables, plane sets, area, metric and topological spaces, and function spaces. 1962 edition. Includes 12 figures. Cover page......Page 1 Title page......Page 4 Preface......Page 6 Table of Contents......Page 9 Part I Functions of a Real Variable......Page 26 Introduction......Page 28 1. An infinite field that cannot be ordered......Page 38 3. An ordered field that is not complete......Page 39 4. A non-Archimedean ordered field......Page 40 6. An ordered field where the rational numbers are not dense......Page 41 8. An integral domain without unique factorization......Page 42 11. Functions continuous on a closed interval and failing to have familiar properties in case the number system is not complete......Page 43 g. A monotonic uniformly continuous nonconstant function having the intermediate value property, and whose derivative is identically 0 on an interval......Page 44 Introduction......Page 45 3. For an arbitrary noncompact set, a continuous and unbounded function having the set as domain......Page 47 6. For an arbitrary noncompact set, a continuous and bounded function having the set as domain and assuming no extreme values......Page 48 8. A bounded function that is nowhere semicontinuous......Page 49 11. A transcendental function......Page 50 13. Two uniformly continuous functions whose product is not uniformly continuous......Page 51 17. A function with a dense set of points of discontinuity every one of which is removable......Page 52 20. A one-to-one correspondence between two intervals that is nowhere monotonic......Page 53 21. A continuous function that is nowhere monotonic......Page 54 23. A function whose points of discontinuity form an arbitrary given F_{\sigma}, set (cf. Example 8, Chapter 4, and Examples 8, 10, and 22, Chapter 8)......Page 55 25. A function with domain [0, 1] whose range for every nondegenerate subinterval of [0, 1] is [0, l] (cf. Example 27, Chapter 8)......Page 56 27. For each n \in R, n(2n+l) functions \phi_{ij}(x_j), j=1,2,...,n, i=l,2,...,2n +l, satisfying: (a) All \phi_{ij}(x_j) are continuous on [0, 1] (b) For any function f(x_1,x_2,...,x_n) continuous for 0 \leq x_1,x_2,...,x_n \leq l, there are 2n+1 functions \psi_i, i=l,2,...,2n +l, each continuous on R, such that f(x_1,x_2,...,x_n) = \sum_{i=1}^{2n+1} \psi_i(\sum_{j=1}^n \phi_{ij}(x_j))......Page 58 1. A function that is not a derivative......Page 60 4. A differentiable function having an extreme value at a point where the derivative does not make a simple change in sign......Page 61 7. A function whose derivative exists and is bounded but possesses no (absolute) extreme values on a closed interval......Page 62 8. A function that is everywhere continuous and nowhere differentiable......Page 63 9. A differentiable function for which the law of the mean fails......Page 64 13. An infinitely differentiable monotonic function f such that \lim_{x \to +\infty} f(x) = O, \lim_{x \to +\infty} f'(x) \neq 0......Page 65 2. A Riemann-integrable function without a primitive......Page 67 6. A function f such that g(x) = \int_0^x f(t)dt is everywhere differentiable with a derivative different from f(x) on a dense set......Page 68 12. A convergent improper integral on [1, +\infty) whose integrand is positive, continuous, and does not approach zero at infinity......Page 70 9. A Riemann-integrable function of a Riemann-integrable function that is not Riemann-integrable (cf. Example 34, Chapter 8)......Page 69 14. Functions f and g such that f is Riemann-Stieltjes integrable with respect to g on both [a, b] and [b, c], but not on [a, c]......Page 71 1. Bounded divergent sequences......Page 72 2. For an arbitrary closed set, a sequence whose set of limit points is that set......Page 73 4. For an arbitrary strictly increasing sequence {\phi_n} -- {\phi(n)} of positive integers, a divergent sequence {a_n} such that \lim_{n \to +\infty} (a_{\phi(n)} - a_n) = 0......Page 74 5. Sequences {a_n} and {b_n} such that \varliminf a_n + \varliminf b_n < \varliminf (a_n + b_n) < \varliminf a_n + \varlimsup b_n < \varlimsup (a_n + b_n) -> 0......Page 105 12. Nonuniformly convergent sequences satisfying any three of the four conditions of Dini's theorem......Page 106 Introduction......Page 108 1. A perfect nowhere dense set......Page 110 2. An uncountable set of measure zero......Page 111 3. A set of measure zero whose difference set contains a neighborhood of the origin......Page 112 4. Perfect nowhere dense sets of positive measure......Page 113 5. A perfect nowhere dense set of irrational numbers......Page 114 7. A set of the second category......Page 115 10. A set A for which there exists no function having A as its set of points of discontinuity......Page 116 11. A nonmeasurable set......Page 117 13. A set A of measure zero such that every real number is a point of condensation of A......Page 119 14. A nowhere dense set A of real numbers and a continuous mapping of A onto the closed unit interval [0, 1]......Page 120 15. A continuous monotonic function with a vanishing derivative almost everywhere......Page 121 18. Two continuous functions that do not differ by a constant but that have everywhere identical derivatives (in the finite or infinite sense)......Page 123 20. A set in [0, 1] of measure zero and category II......Page 124 22. A set of measure zero such that there is no function-Riemann--integrable or not--having the set as its set of points of discontinuity......Page 125 24. Two disjoint nonempty nowhere dense sets of real numbers such that every point of each set is a limit point of the other......Page 126 25. Two homeomorphic sets of real numbers that are of different category......Page 127 26. Two homeomorphic sets of real numbers such that one is dense and the other is nowhere dense......Page 128 27. A function defined on R, equal to zero almost everywhere and whose range on every nonempty open interval is R......Page 129 31. A bounded semicontinuous function that is not Riemann-integrable, nor equivalent to a Riemann-integrable function......Page 130 34. A Riemann-integrable function f, and a continuous function g, both defined on [0, 1] and such that the composite function f(g(x)) is not Riemann-integrable on [0, 1], nor equivalent to a Riemann-integrable function there (cf. Example 9, Chapter 4)......Page 131 35. A bounded function possessing a Primitive on a closed interval but failing to be Riemann-integrable there......Page 132 37. A function that is Lebesgue-measurable but not Borel-measurable......Page 133 40. Sequences of functions converging in different senses......Page 134 41. Two measures \mu and \nu on a measure space (X, S) such that \mu is absolutely continuous with respect to \nu and for which no function f exists such that \mu(E) = \int_E f(x)d\nu(x) for all E \in S......Page 137 Part II. Higher Dimensions......Page 138 1. A discontinuous function of two variables that is continuous in each variable separately......Page 140 3. A refinement of the preceding example......Page 141 6. Functions f for which exactly one of the following exists: \lim_{(x,y) \to (a,b)} f(x,y), \lim_{x \to a} \lim_{y \to b} f(x,y), \lim_{y \to b} \lim_{x \to a} f(x,y)......Page 142 8. A function fix, y) for which \lim_{y \to 0} f(x,y) = g(x) exists uniformly in x, \lim_{x \to 0} f(x,y) = h(y) exists uniformly in y, \lim_{x \to 0} g(x) = \lim_{y \to 0} h(y) g(x) but \lim_{(x,y) \to (0,0)} f(x,y) does not exist......Page 143 9. A differentiable function of two variables that is not continuously differentiable......Page 144 10. A differentiable function with unequal mixed second-order partial derivatives......Page 145 12. A locally homogeneous continuously differentiable function of two variables that is not homogeneous......Page 146 14. A refinement of the preceding example......Page 147 15. A function f for which \frac{d}{dx} \int_a^b f(x, y) dy \neq \int_a^b \left[ \frac{\partial}{\partial x} f(x, y) \right ] dy, although each integral is proper......Page 148 17. A double series \sum_{m,n} a_{mn} for which \sum_m \sum_n a_{mn} \neq \sum_n \sum_m a_{mn} although convergence holds throughout......Page 149 18. A differential P dx + Q dy and a plane region R in which P dx + Q dy is locally exact but not exact......Page 150 19. A solenoidal vector field defined in a simply-connected region and possessing no vector potential......Page 151 Introduction......Page 153 2. A bounded plane set contained in no minimum closed disk......Page 155 3. "Thin" connected sets that are not simple arcs......Page 156 5. A mapping of the interval [0, 1] onto the square [0, 1] × [0, l]......Page 157 6. A space-filling arc in the plane......Page 158 9. A continuous mapping of [0, 1] onto [0, 1] that assumes every value an uncountable number of times......Page 159 10. A simple arc in the unit square and of plane measure arbitrarily near 1......Page 160 13. Three disjoint plane regions with a common frontier......Page 163 14. A non-Jordan region equal to the interior of its closure......Page 164 17. A simple arc of infinite length and having a tangent line at every point......Page 165 19. A smooth curve C containing a point P that is never the nearest point of C to any point on the concave side of C......Page 166 21. A nonmeasurable plane set having at most two points in common with any line......Page 167 22. A nonnegative function of two variables f(x, y) such that \int_0^1 \int_0^1 f(x, y) dxdy = \int_0^1 \int_0^1 f(x, y) dydx = 0 and such that \int \int_S f(x, y) dA, where $ = [0, 1] × [0, 1], does not exist......Page 169 23. A real-valued function of one real variable whose graph is a nonmeasurable plane set......Page 170 24. A connected set that becomes totally disconnected upon the removal of a single point......Page 171 Introduction......Page 172 2. A compact plane set without area......Page 173 6. Two functions \phi and \psi defined on [0, 1] and such that (a) \phi(x) < \psi(x) for x \in [0, 1]l (b) \int_0^1 [\phi(x) - \psi(x)]dx exists and is equal to 1; (c) S \equiv \{ (x, y) | 0 \leq x \leq 1,\phi(x) < y 0, two Euclidean balls B_{\epsilon} and B_{M} of radius \epsilon and M respectively, a decomposition of B_{\epsilon} into a finite number of disjoint subset S_1, S_2, ..., S_n, and n rigid motions R_1, R_2, ..., R_n, such that B_M = \cong R_1{S_1) \cup R_2{S_2) \cup ... \cup R_n{S_n)......Page 196 Introduction......Page 197 3. Two semicontinuous functions whose sum is not semicontinuous......Page 200 5. Two functions whose squares are Lebesgue-integrable and the square of whose sum is not Lebesgue-integrable......Page 202 9. Two metrics for the space C([0, 1]) of functions continuous on [0, 1] such that the complement of the unit ball in one is dense in the unit ball of the other......Page 203 Bibliography......Page 205 Special Symbols......Page 208 Index......Page 212