Entrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems

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1 September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion. 2. Let f n (x) = nx n + x. (a) Find f(x) = f n (x); (b) Does f n converge to f uniformly (, )? Prove your conclusion. (c) Does f n converge to f uniformly (, )? Prove your conclusion. 3. If s n, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ > such that N >, n > N with s n. (d) s2 n ( s n) Let g(x) be dened as follows g(x) = (a) Prove g(x) is continuous in (, ). cos(xy 3 ) + y 2 dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Dene f by f(x) = n= a nx n. (a) Show that f is a function of bounded variation on x [, ]. (b) If a and a n = for n 2, is f absolutely continuous on x R. Explain your answer. (c) If a, a 2, and a n = for n 3, is f absolutely continuous on x R. Explain your answer.

2 6. For p < dene f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. Either prove the statement or show a counter example. (a) If E = [, ], then there is a constant c > for which f c f 2 for all f L 2 (E). (b) Suppose E = [, ). L (E). If f is bounded and E f 2 <, then it belongs to 7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R. Show that f(x) =. x 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =. 2

3 April 2, 27 Solve exactly 6 out of the 8 problems. Is f(x) = x 3 uniformly continuous in (-5, )? in (, )? Prove your conclusion. 2. Let f n (x) = nx n + x. (a) Find f(x) = f n (x); (b) Does f n converge to f uniformly (, )? Prove your conclusion. (c) Does f n converge to f uniformly (, )? Prove your conclusion. 3. If s n, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ > such that N >, n > N with s n > ɛ. (d) s2 n > ( s n) Let g(x) be dened as follows g(x) = (a) Prove g(x) is continuous in (, ). cos 2 (xy) + y 2 dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Dene f : [, ] R by f(x) = n= a nx n. Show that f is a function of bounded variation. 6. Suppose that f : [, ) [, ) is measurable and that f(x) <. Prove that x n f(x) + x n dx = f(x)dx 7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R. Show that f(x) =. x 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =.

4 August 3, 26 Solve exactly 6 out of the 8 problems. (a) Prove by denition that x 2 is uniformly continuous in ( 7, 2). (b) Prove by denition that x 2 is not uniformly continuous in [, ). 2. If s n =, which of the following statements are true? Explain your conclusion. (a) N > such that n > N, s n. (b) N >, n > N such that s n. (c) ɛ >, N >, such that n > N, s n > ɛ. (d) ɛ > such that N >, n > N with s n > ɛ. 3. Let f(x) be a function of bounded variation dened on [,]. Prove that the set of all discontinuity points of f(x) is countable. 4. Let f(x) be a measurable function dened in R and satisfying: (a) f() = ; (b) If f(x ) >, then δ > such thatf(x) > in (x δ, x + δ); (c) If f(x n ) > and x n a, then f(a) > ). Is it true f(x) > x R? Prove your conclusion. 5. Suppose that {f n } is a sequence of real valued measurable functions dened on the interval [, ] and suppose that f n(x) = f(x) a.e. on [, ]. Let p >, M >, and that f n L p (R) M for all n. (a) Prove that f L p (R) and that f L p (R) M. (b) Prove that f f n L p (R) =. 6. Suppose that f L (R) is a uniformly continuous function. Show that f(x) =. x

5 7. For p < dene f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. statement below, either prove it or show a counter example. For each (a) If E = [, ], then there is a constant c for which f 2 c f for all f L (E). (b) Suppose E = [, ). If f is bounded and E f 2 <, then it belongs to L (E). 8. (a) For a Lebesgue integrable function f on [,] dene F (x) = x f(t)dt for x [, ]. Prove that F is absolutely continuous on [,]. (b) Give an explicit example of a continuous function F of bounded variation on [,] that is not absolutely continuous on [,]. It is not necessary to verify your assertion. 2

6 April 2, 26 Solve exactly 6 out of the 8 problems. Let f be a monotone function dened on [,] with f() = and f() =. Show that f has at most countable discontinuous points. 2. Answer the following questions and prove your conclusion. (a) Is x 2 uniformly continuous in the open interval ( 5, 2)? (b) Is x 2 uniformly continuous in [, )? 3. If s n >, which of the following statements are true? Prove your conclusion. (a) N > such that n > N, s n >. (b) ɛ > such that N >, n > N with s n > ɛ. (c) ɛ >, N >, such that n > N, s n > ɛ. (d) s2 n = ( s n) Let f L (, ), and g(x) be dened as follows g(x) = ˆ (a) Prove g(x) is continuous in (, ). cos 2 (xy)f(y)dy. (b) Is g(x) uniformly continuous in (, )? absolutely continuous in (, )? Prove your conclusion. 5. Let f and g be real valued measurable functions on [, ] with the property that g is dierentiable at every x on [, ] and Show that f L [, ]. g (x) = (f(x)) (a) Let m be the Lebesgue measure on R and let E R be measurable. Also, suppose f L (E) and that f > a.e. on E. Show that ˆ f(x) /n dx = m(e). E (b) Let E = [a, b], where a and b are real numbers satisfying a < b. With the same assumptions on f as in (a), show that and (ˆ b f(x) /n dx) n = if b a > a (ˆ b f(x) /n dx) n = if b a <. a

7 7. Suppose that f : [, ) [, ) is measurable and that f(x) <. Prove that ˆ x n ˆ f(x) + x n dx = f(x)dx 8. Let {u n } n= be a sequence of Lebesgue measurable functions on [, ] and assume u n (x) = a.e. on [, ], and also u n L 2 [,] for all n. Prove that u n L [,] =. 2

8 April 7, 24 Solve exactly 6 out of the 8 problems.. Answer the following questions and prove your conclusion. If A is a closed subset of [,] and A [, ]. Is it possible that ma =? If B is an open subset of [,] and dense in [, ]. Is it possible that mb <? 2. Let {p k } be all the rational numbers in [,], and H(x) be the Heaviside function defined as {, x >, H(x) =, x. Define f(x) in [,] as f(x) = ( ) k 2 k H(x p k ). k= Prove by definition (in ɛ δ language) that f(x) is continuous at all irrational numbers, and discontinuous at all rational numbers in [,]. Determine and explain: (i). Is f(x) Riemann integrable in [,]? (ii). Is f(x) Lebegue integrable? (iii) Is f(x) of bounded total variation? 3. Let f be Lebegue integrable in [,b] with b <. Prove b f(x) cos nx dx =. Is it also true if b =? Prove your conclusion. 4. If f is absolutely continuous on [,] and f(x) > on [,]. Prove by definition that /f is absolutely continuous on [,]. Is f(x) = x absolutely continuous on [,]? In [, )? Prove your conclusion. 5. Prove or disprove that for every ε > and for every f L [a, b], where a and b are finite, there is a g C[a, b] such that 6. Compute the Lebesgue integral ess sup a x b f g < ε. inf xex sin 2 (πnx) dx.

9 7. Prove or disprove the following statement. (a) Let g be an integrable function on [; ]. Then there is a bounded measurable function f such that fg = g f. (b) Let {f n } be a sequence of functions in L [, ], which converge almost everywhere to a function f in L, and suppose that there is a constant M such that f n M all n. Then for each function g in L we have fg = 8. Let {a n } n= be a sequence of real numbers satisfying n= a n <. Define f : [, ] R by f(x) = n= a2 nx n. Show that f is a function of bounded variation in [, ]. f n g. 2

10 April 22, 25 Solve exactly 6 out of the 8 problems.. Let A R be an open set with the following property: {x n } A, a subsequence {x nk } such that x nk x A. Which of the following statements is true? ma =? ma = ln 2? ma = π? ma =? ma is not uniquely determined and could be any positive number? Prove your conclusion. 2. Answer the following questions and prove your conclusion. Let A [, ] be an open set. If δ < such that ma δ, is it possible that A is dense in [,]? Let B [, ] and B [, ]. If ɛ >, mb > ɛ, is it possible that B is closed? 3. Let f L (, ), and g(x) be defined as follows g(x) = ˆ Prove g(x) is continuous in (, ). cos(xy)f(y)dy. Is g(x) uniformly continuous in (, )? Prove your conclusion. 4. Let f(x) > and be absolutely continuous on [,]. Let g(x) = is absolutely continuous on [,]. f(x). Prove g(x) 5. (a) For a bounded Lebesgue integrable function f on [,] define F (x) = x f(t)dt for x [, ]. Prove that F is absolutely continuous on [,]. (b) Give an explicit example of a continuous function F of bounded variation on [,] that is not absolutely continuous on [,]. It is not necessary to verify your assertion.

11 6. (a) Suppose that f L [, ] and let m denote Lebesgue measure. Prove that for c > m{ f(x) c} c ˆ f(x) dx. (b) Suppose that {f n } L p [, ] is a sequence of functions satisfying f n L p [,] M, where < p <. Prove that ˆ f n (x) dx =, uniformly in n. c f n(x) c 7. Assume that n is an integer and let f n= Ln [, ]. Prove that if then f = a.e. f L n [,] <, n= 8. Suppose that {f n } is a sequence in L (R) with f n L (R) for all n and f n(x) = f(x) a.e. (a) Prove that f L (R) and that f L (R). (b) Show that ( f f n L (R) f n L (R) + f L (R) ) =. Hint: The following inequality might be useful. For any numbers a and b a b a + b 2 b 2

12 August 27, 23 Solve exactly 6 out of the 8 problems. Given function φ(x) with φ() = and φ (x) = e x2. The plane curve Γ is dened by the equation y = φ( + xy + y 2 ). Find the equation for the tangent line to the curve Γ at the point (x, y) = (, ). 2. Does ( ) x f n (x) = x n sin x converge uniformly on (,)? Prove your conclusion. 3. Let {x n = p n /q n Q} be a sequence of rational numbers and x n α R\Q (α is irrational). Prove q n. 4. Let E be a Lebesgue measurable set in R, and f n be a sequence of nonnegative Lebesgue integrable functions on E. If f n f a.e. on E, is it always true that ˆ ˆ fdx f n dx? Prove your conclusion. E 5. Let a and b are real numbers satisfying a < b. Prove or disprove the following. (a) If f(x) is a dierentiable function on a < x < b, it is absolutely continuous on a < x < b. (b) If f(x) is absolutely continuous on a x b, it is Lipschitz on a x b. 6. A family F of measurable functions on E of nite measure is said to be uniformly integrable over E provided that for each ε >, there is a δ > such that for each f F and for each measurable set A E with ma < δ, A f < ε. Either prove or disprove the following statements. (a) If G is the family of measurable functions f on [, ], each of which is integrable over [, ] and has f <, then G is uniformly integrable over [, ]. (b) If {f n } is a sequence of nonnegative, measurable, and integrable functions that converges pointwise a.e. on E to h =, then {f n } is uniformly integrable. E

13 7. For p < dene ˆ f p = ( f p ) /p E and denote L p (E) to be the set of functions f for which E f p <. Either prove the statement or show a counter example. (a) If E = [, ], then there is a constant c for which f 2 c f for all f L (E). (b) Suppose E = [, ). If f is bounded and E f 2 <, then it belongs to L (E). 8. Prove or disprove that for every ε > and for every f L [a, b], where a and b are nite, there is a g C[a, b] such that ess sup a x b f g < ε. 2

14 April 22, 23 Solve exactly 6 out of the 8 problems. Given function φ(x) with φ() = 5 and φ (x) = e x2. The plane curve Γ is dened by the equation y = φ( + sin 3(xy)). Find the equation for the tangent line to the curve Γ at the point (x, y) = (, 5). 2. Does ( ) x f n (x) = x n sin x converge uniformly on (,)? Prove your conclusion. 3. Evaluate the integral I = ˆ dy ˆ y e x2 dx. 4. Let f, g be absolutely continuous in (,). Is it true that the product fg is also absolutely continuous in (,)? Prove your conclusion. 5. Let {f n } be a sequence of nonnegative integrable functions on E. For each of (a), (b), and (c) either prove the statement or show a counter example. (a) If E f n =, then {f n } in measure. (b) If {f n } in measure, then E f n =. (c) If {f n } almost everywhere on E, then E f n =. 6. For f in C(E), where C(E) is the set of continuous functions on E, dene ˆ f p = ( f p ) /p, f sup = sup f(x). x E E Here p <. For each of (a), (b), and (c) either prove the statement or show a counter example. (a) If E = [, ] and p < q <, then there is a constant c for which f p c f q for all f C(E). (b) If E = [, ], then there is a constant c for which f sup c f for all f C(E). (c) Suppose E = [, ). If f C(E) is bounded and E f 2 <, then it belongs to L (E).

15 7. Prove or disprove that for every ε > and for every f L [a, b], there is a g C[a, b] such that ess sup a x b f g < ε. 8. (a) Suppose E R has Lebesgue measure zero. Prove that if f : R R is increasing and absolutely continuous, then f(e) has Lebesgue measure zero. Here, f(e) = {y y = f(x) for some member x of E}. (b) Would the same conclusion hold if we remove increasing in the assumption on f? In other words, if f : R R is absolutely continuous, then would f(e) have Lebesgue measure zero? Explain your reasoning. 2