Real Analysis Problems Cristian E. Guti´errez September 14, 2009 1 1 CONTINUITY 1 Continuity Problem 1.1 Let rn be the sequence of rational numbers and X 1 f(x) = : 2n fn:rn<xg Prove that 1. f is continuous on the irrationals. 2. f is discontinuous on the rationals. R 1 3. Calculate 0 f(x) dx: 1 Hint: for (3) set A(x) = fn : r < xg so f(x) = P1 χ (n) ; use Fubini. n n=1 A(x) 2n p 2 2 Problem 1.2 Let fn(x) = sin x + 4 n π on [0; +1): Prove that 1. fn is equicontinuous on [0; +1): 2. fn is uniformly bounded. 3. fn ! 0 pointwise on [0; +1): 4. There is no subsequence of fn that converges to 0 uniformly. 5. Compare with Arzel`a-Ascoli. Problem 1.3 Prove that the class of Lipschitz functions f in [a; b] with Lipschitz constant ≤ K and f(a) = 0 is a compact set in C([a; b]): Problem 1.4 Let B be the unit ball in C([a; b]). Define for f 2 C([a; b]) b Z 2 T f(x) = −x2 + e−x +y f(y) dy: a Prove that T (B) is relatively compact in C[a; b]: 2 1 CONTINUITY Problem 1.5 Let gn : R ! R be the continuous function that is zero outside the interval [0; 2=n], gn(1=n) = 1, and gn is linear on (0; 1=n) and (1=n; 2=n): Prove that gn ! 0 pointwise in R but the convergence is not uniform on any interval containing 0. Let rk be the rational numbers and define 1 X −k fn(t) = 2 gn(t − rk): k=1 Prove that 1. fn is continuous on R. 2. fn ! 0 pointwise in R. 3. fn does not converge uniformly on any interval of R. n−1 1 X k Problem 1.6 Let f 2 C( ) and f (x) = f x + . Show that f con- R n n n n k=0 verges uniformly on every finite interval. Problem 1.7 Consider the following statements: (a) f is a continuous function a.e. on [0; 1] (b) there exists g continuous on [0; 1] such that g = f a.e. Show that (a) does not imply (b), and (b) does not imply (a). Problem 1.8 Show that if f and g are absolutely continuous functions in [a; b] and f 0(x) = g0(x) a.e., then f(x) − g(x) = constant, for each x 2 [a; b]: Problem 1.9 If f is absolutely continuous in [0; 1], then f 2 is absolutely continuous in [0; 1]. Problem 1.10 Let 0 < α ≤ 1. A function f 2 Cα([0; 1]), f is H¨oldercontinuous of order α, if there exists K ≥ 0 such that jf(x)−f(y)j ≤ K jx−yjα for all x; y 2 [0; 1]: Prove that 1. Cα[0; 1] ⊂ Cβ([0; 1]) for 0 < β ≤ α ≤ 1: 2. If f is H¨oldercontinuous of order one, that is, f is Lipschitz, then f is abso- lutely continuous on [0; 1]: 3 1 CONTINUITY 3. Let a > 0. Define g(x) = xa sin(x−a) for x 6= 0 and f(0) = 0. 4. Prove that g2 = BV [0; 1]. Prove that g 2 Cα([0; 1]) with α = a=(1 + a), and g2 = Cβ([0; 1]) for β > a=(1 + a). Problem 1.11 Consider the space BV [a; b] of bounded variation functions in [a; b] with the norm k X kfkBV = sup jf(αi) − f(αi−1)j + jf(a)j; Γ i=1 where the supremum is taken over all partitions Γ = fα0; : : : ; αkg of [a; b]. Prove that 1. (BV [a; b]; k · k) is a Banach space; 2. the set of absolutely continuous functions in [a; b] is a closed subspace of BV [a; b]. Problem 1.12 Show that the series 1 X (−1)k k + jxj k=1 converges for each x 2 R and the sum is a Lipschitz function. Problem 1.13 Let f :[a; b] ! R be continuous and fxkg ⊂ [a; b] a Cauchy se- quence. Prove that f(xk) is a Cauchy sequence. Problem 1.14 Let f : R ! R be such that jf(x) − f(y)j ≤ M jx − yjα for some M > 0, α > 1 and for all x; y 2 R. Prove that f is constant. Problem 1.15 Let f(x) = x2 sin(1=x2) for x 2 [−1; 1], x 6= 0, and f(0) = 0: Show that f is differentiable on [−1; 1] but f 0 is unbounded on [−1; 1]: Problem 1.16 Suppose fn ! f uniformly in Ω open, and fxng ⊂ Ω with xn ! x 2 Ω. Prove that fn(xn) ! f(x). n Problem 1.17 If xn is a sequence such that jxn+1 − xnj ≤ 1=2 , then fxng is a Cauchy sequence. 4 1 CONTINUITY Problem 1.18 Let a < b; c < d: Define 8 ax sin2 1 + bx cos2 1 ; for x > 0 <> x x f(x) = 0; for x = 0 :> 2 1 2 1 cx sin x + dx cos x ; for x > 0: − + Calculate D f; D f; D−f; D+f at x = 0: Problem 1.19 Let f be a continuous function in [-1,2]. Given 0 ≤ x ≤ 1; and n ≥ 1 define the sequence of functions x+ 1 n Z n fn(x) = f(t) dt: 2 1 x− n Show that fn is continuous in [0,1] and fn converges uniformly to f in [0,1]. Problem 1.20 Let fn : R ! R be a uniformly bounded sequence of functions. Show that for each countable subset S ⊂ R there exists a subsequence of fn which converges in S. Hint: select the subsequence by using a diagonal process Problem 1.21 Let f be a continuous function in [0; 1] such that f is absolutely continuous in [0; ] for every , 0 < < 1: Show that f is absolutely continuous in [0; 1]: Problem 1.22 Let fn be absolutely continuous functions in [a; b], fn(a) = 0: Sup- 0 1 pose fn is a Cauchy sequence in L ([a; b]). Show that there exists f absolutely continuous in [a; b] such that fn ! f uniformly in [a; b]: Problem 1.23 Let fn(x) = cos(n x) on R. Prove that there is no subsequence fnk converging uniformly in R. Problem 1.24 Let f : Rn ! R be a bounded function. If E ⊂ Rn, then the oscillation of f over E is defined by oscEf = sup f − inf f; E E n and for x 2 R define !f (x) = limδ!0 oscBδ(x)f: Prove 1. If E ⊂ F , then oscEf ≤ oscF f. 2. Prove that !f is well defined and !f (x) = infδ>0 oscBδ(x)f: 3. For each α > 0 the set fx : !f (x) < αg is open. 4. The set of points of discontinuity of f is an Fσ. 5 2 SEMICONTINUOUS FUNCTIONS 2 Semicontinuous functions Definition 2.1 Let Ω ⊂ Rn be open. The function f :Ω ! R is lower (up- per) semicontinuous if f(z) ≤ lim infx!z f(x) = limδ!0 infjx−zj<δ f(x) (f(z) ≥ lim supx!z f(x) = limδ!0 supjx−zj<δ f(x)). Problem 2.2 Prove that the following statements are equivalent: 1. f is lower semicontinuous; 2. f −1(c; +1) is open for each c 2 R; 3. f −1(−∞; c] is closed for each c 2 R; 4. If xk ! z, then f(z) ≤ supk f(xk); 5. For each z 2 Ω and for each > 0 there exists a neighborhood V of z such that f(x) ≥ f(z) − for each x 2 V: Problem 2.3 Let g :Ω ! R and define f(x) = lim inf g(z): z!x Prove that f is lower semicontinuous. Problem 2.4 Prove that n 1. if E ⊂ R , then χE is lower semicontinuous if and only if E is open. 2. the function 8 <0 if x is irrational f(x) = 1 if x = m=n with n 2 , m 2 , and m=n irreducible :n N Z is upper semicontinuous. Problem 2.5 Let ffαgα2J be a family of lower semicontinuous function in Ω: Prove that f(x) = supα2J fα(x) is lower semicontinuous. Problem 2.6 Prove that the function defined in Problem 1.1 is lower semicontin- uous in R. 6 3 MEASURE 3 Measure n Problem 3.1 Let Ej be a sequence of sets in R and E = lim supj!1 Ej = T1 S1 k=1 j=k Ej: Show that lim supj!1 χEj (x) = χE(x): 1 Problem 3.2 Consider the set of rational numbers Q = fqjgj=1: Prove that 1 [ 1 1 n q − ; q + 6= ;: R j j2 j j2 j=1 Problem 3.3 Construct a countable family of closed intervals contained in [0; 1] such that the union covers [0; 1] but there is no finite subcovering. n Problem 3.4 Let E be a set in R . Show that there exists a sequence Gi of open sets, G1 ⊃ G2 ⊃ · · · ⊃ E such that 1 \ j Gij = jEje: i=1 Problem 3.5 If E ⊂ [0; 1], jEj = 0 and f(x) = x3, then jf(E)j = 0: Problem 3.6 True or false, justify your answer. 1. The class of Lebesgue measurable sets has cardinality 2c. 2. Every perfect set has positive measure.