real analysis problems - cristian e. gutierrez

8/14/2019 Real Analysis Problems - Cristian E. Gutierrez

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Real Analysis Problems

Cristian E. Gutierrez

September 14, 2009

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1 CONTINUITY

1 ContinuityProblem 1.1 Let r n be the sequence of rational numbers and

f (x) ={n :r n


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1 CONTINUITY

Problem 1.5 Let gn : R R be the continuous function that is zero outside theinterval [0, 2/n ], gn (1/n ) = 1, and gn is linear on (0, 1/n ) and (1/n, 2/n ). Prove thatgn 0 pointwise in R but the convergence is not uniform on any interval containing0. Let r k be the rational numbers and dene

f n (t) =

k=1

2 k gn (t r k).

Prove that

1. f n is continuous on R .

2. f n

0 pointwise in R .

3. f n does not converge uniformly on any interval of R .

Problem 1.6 Let f C (R ) and f n (x) = 1n

n 1

k=0

f x + kn

. Show that f n con-

verges uniformly on every nite 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] andf (x) = g (x) a.e., then f (x) g(x) = constant, for each x [a, b].

Problem 1.9 If f is absolutely continuous in [0, 1], then f 2 is absolutely continuousin [0, 1].

Problem 1.10 Let 0 < 1. A function f C ([0, 1]), f is H older continuous of order , if there exists K 0 such that |f (x)f (y)| K |xy| for all x, y [0, 1].Prove that1. C [0, 1] C ([0, 1]) for 0 < 1.2. If f is Holder continuous of order one, that is, f is Lipschitz, then f is abso-

lutely continuous on [0, 1].

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1 CONTINUITY

3. Let a > 0. Dene g(x) = xa sin(x a ) for x = 0 and f (0) = 0.4. Prove that g / BV [0, 1]. Prove that g C ([0, 1]) with = a/ (1 + a), andg / C ([0, 1]) for > a/ (1 + a).

Problem 1.11 Consider the space BV [a, b] of bounded variation functions in [a, b]with the norm

f BV = sup

k

i=1|f ( i) f ( i 1)|+ |f (a)|,

where the supremum is taken over all partitions = {0, . . . , k} of [a, b]. Provethat1. (BV [a, b], ) 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

k=1

(1)kk + |x|

converges for each x R and the sum is a Lipschitz function.

Problem 1.13 Let f : [a, b] R be continuous and {xk} [a, b] a Cauchy se-quence. Prove that f (xk) is a Cauchy sequence.

Problem 1.14 Let f : R R be such that |f (x) f (y)| M |x y| for someM > 0, > 1 and for all x, y R . Prove that f is constant.

Problem 1.15 Let f (x) = x2 sin(1/x 2) for x [1, 1], x = 0, and f (0) = 0 . Showthat f is differentiable on [1, 1] but f is unbounded on [1, 1].

Problem 1.16 Suppose f n f uniformly in open, and {xn} with xn x . Prove that f n (xn ) f (x).

Problem 1.17 If xn is a sequence such that |xn +1 xn | 1/ 2n , then {xn} is aCauchy sequence.

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1 CONTINUITY

Problem 1.18 Let a < b,c < d. Dene

f (x) =ax sin2 1x + bxcos2 1x , for x > 00, for x = 0cx sin2 1x + dx cos

2 1x , 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, andn 1 dene the sequence of functionsf n (x) =

n2

x+ 1n

x 1n

f (t) dt.

Show that f n is continuous in [0,1] and f n converges uniformly to f in [0,1].

Problem 1.20 Let f n : R R be a uniformly bounded sequence of functions.Show that for each countable subset S R there exists a subsequence of f n whichconverges 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 absolutelycontinuous in [0, ] for every , 0 < < 1. Show that f is absolutely continuous in[0, 1].

Problem 1.22 Let f n be absolutely continuous functions in [ a, b], f n (a) = 0 . Sup-pose f n is a Cauchy sequence in L1([a, b]). Show that there exists f absolutelycontinuous in [a, b] such that f n f uniformly in [a, b].Problem 1.23 Let f n (x) = cos( n x ) on R . Prove that there is no subsequence f n kconverging uniformly in R .

Problem 1.24 Let f : Rn R be a bounded function. If E Rn , then theoscillation of f over E is dened byoscE f = sup

E f

inf E

f,

and for x R n dene f (x) = lim 0 oscB (x)f. Prove1. If E F , then oscE f oscF f .2. Prove that f is well dened and f (x) = inf > 0 oscB (x)f.

3. For each > 0 the set {x : f (x) < } is open.4. The set of points of discontinuity of f is an F .

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2 SEMICONTINUOUS FUNCTIONS

2 Semicontinuous functionsDenition 2.1 Let Rn be open. The function f : R is lower (up-per) semicontinuous if f (z ) lim inf xz f (x) = l im 0 inf |x z |


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3 MEASURE

3 MeasureProblem 3.1 Let E j be a sequence of sets in Rn and E = lim sup j E j =

k=1

j = k E j . Show that lim sup j E j (x) = E (x).

Problem 3.2 Consider the set of rational numbers Q = {q j } j =1 . Prove that

R \

j =1

q j 1 j 2

, q j + 1 j 2

= .

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 nite subcovering.

Problem 3.4 Let E be a set in Rn . Show that there exists a sequence Gi of opensets, G1 G2 E such that

|

i=1

Gi| = |E |e.

Problem 3.5 If E

[0, 1],

|E

| = 0 and f (x) = x3, then

|f (E )

| = 0.

Problem 3.6 True or false, justify your answer.

1. The class of Lebesgue measurable sets has cardinality 2 c.

2. Every perfect set has positive measure.

3. Every bounded function is measurable.

Problem 3.7 Let be an exterior measure and let An be a sequence of sets suchthat

n =1

(An ) < + .Show that (lim An ) = 0 . ( lim An = n =1

j = n A j )

Problem 3.8 Let f : [0, 1] [0, 1] be the Cantor function, and let be theLebesgue-Stieltjes measure generated by f , i.e. ((a, b]) = f (b) f (a), for (a, b] [0, 1]. Show that is singular with respect to Lebesgue measure.

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3 MEASURE

Problem 3.9 Let be an uncountable set. The class of sets dened by

F = {A : A is countable or Ac is countable}is a -algebra. Dene the measures: (A) = + if A is innite, (A) = #( A) if Ais nite; (A) = 0 if A is countable and (A) = 1 if A is uncountable. Show that but an integral representation of the form (A) = A f d is not valid. Whythis does not contradict the Radon-Nikodym theorem?Problem 3.10 Show that given , 0 < < 1, there exists a set E [0, 1] which isperfect, nowhere dense and |E | = 1 .Hint: the construction is similar to the construction of the Cantor set, exceptthat at the kth stage each interval removed has length 3 k .

Problem 3.11 Let {I 1, , I N } be a nite family of intervals in R such that Q [0, 1] N j =1 I j . Prove that N j =1 |I j | 1. Is this true if the family of intervals isinnite?

Problem 3.12 Let {E j} j =1 be a sequence of measurable sets in R n such that |E j E i| = 0 for j = i. Prove that

|

j =1

E j | =

j =1|E j |.

Problem 3.13 Let A, B Rn such that A is measurable. If A B = then|A B |e = |A|+ |B |e .Hint: given > 0 there exists F closed such that F A and |A \ F | < .

Problem 3.14 Let {E j} j =1 be a sequence of measurable sets in Rn . Prove that1. liminf j E j and limsup j E j are measurable.

2. |liminf j E j | lim inf j |E j |.3. If | j k E j | < for some k then |limsup j E j | lim sup j |E j |.

Problem 3.15 Let A R measurable with |A| < . Show that the functionf (x) = |(, x) A| is nondecreasing, bounded and uniformly continuous in R .

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3 MEASURE

Problem 3.16 Show that Q is not a set of type G. Construct a set G of type G

such that Q G and |G| = 0.Problem 3.17 Let E be a measurable set in R with positive measure. We say thatx R is a point of positive measure with respect to E if |E I | > 0 for each open in-terval I containing x. Let E + = {x R : x is of positive measure with respect to E }.Prove that

1. E + is perfect.

2. |E \E + | = 0.

Problem 3.18 Let E be the set of numbers in [0, 1] whose binary expansion has 0in all the even places. Show that |E | = 0.

Problem 3.19 Let f k be measurable and f k f a.e. in Rn . Prove that thereexists a sequence of measurable sets {E j } j =1 such that |R n \ j =1 E j | = 0 and f k f uniformly on each E j .

Problem 3.20 Let E Rn be a measurable set with positive measure, and letD R

n

be a countable dense set. Prove that |Rn

\ qk D (q k + E )| = 0.Problem 3.21 An ellipsoid in Rn with center at x0 is a set of the form

E = {x R n : A(x x0), x x0 1},where A is an nn positive denite symmetric matrix and , denotes the Euclideaninner product. Prove that

|E | = n

(det A)1/ 2,

where n is the volume of the unit ball.Hint: formula of change of variables.

Problem 3.22 Let f : R R be a continuously differentiable function. Dene : R 2 R 2 by (x, y) = ( x + f (x + y), y f (x + y)).Prove that |(E )| = |E | for each measurable set E R 2.Hint: formula of change of variables.

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3 MEASURE

Problem 3.23 Let E [0, 1] such that there exists > 0 satisfying |E [a, b]| (ba) for all [a, b] [0, 1]. Prove that |E | = 1.Problem 3.24 We say that the sets A, B Rn are congruent if A = z + B forsome z R n .Let E R n be measurable such that 0 < |E | < + . Suppose that there existsa sequence of disjoint sets {E i}i=1 such that for all i, j , E i and E j are congruent,and E = j =1 E j .Prove that all the E j s are nonmeasurable.

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4 MEASURABLE FUNCTIONS

4 Measurable functionsLet E be a measurable set and f n : E R a sequence of measurable functions.(A) f n f pointwise in E if f n (x) f (x) for all x E .(B) f n f pointwise almost everywhere in E if f n (x) f (x) for almost all x E .(C) f n f almost uniformly in E if for each > 0 there exists a measurable setF E such that |E \F | and f n f uniformly in F .(D) f n f in measure in E if for each > 0, limn |{x E : |f n (x) f (x)|

}| = 0.(E) f n is a Cauchy sequence in measure in E if for each , > 0, there exists N

such that |{x E : |f n (x) f m (x)| }| for all n, m N.f : E R is a measurable function if f 1((, a)) is a Lebesgue measurableset for all a R .f : E R is a Borel measurable function if f 1((, a)) is a Borel set for alla R .

Problem 4.1 If f n f almost uniformly in E , then f n f in measure in E .

Problem 4.2 If f n f in measure and f n g in measure, then f = g a.e.

Problem 4.3 If f n is a Cauchy sequence in measure, then there exists a subsequencef n k that is a Cauchy sequence almost uniformly.

Problem 4.4 If f n is a Cauchy sequence in measure, then there exists f measurablesuch that f n f in measure.

Problem 4.5 Give an example of a sequence f n that converges in measure but doesnot converge a.e.

Problem 4.6 If |E | < and f n f a.e. in E , then f n f in measure. Showthat this is false if |E | = .

Problem 4.7 If f n f in measure, then there exists a subsequence f n k such thatf n k f a.e.

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4 MEASURABLE FUNCTIONS

Problem 4.8 If f n f in measure, then f n is a Cauchy sequence in measure.

Problem 4.9 Let f , f k : E R be measurable functions. Prove that1. If f k f in measure, then any subsequence f k j contains a subsequence f kj m f a.e. as m , .2. Suppose |E | < . If any subsequence f kj contains a subsequence f k j m f a.e. as m , then f k f in measure.

Hint: for (2) suppose by contradiction that f k f in measure. Then there exists0 > 0 such that

|{x

E :

|f k(x)

f (x)

| 0

}| 0. Hence there is an increasing

sequence k j such that |{x E : |f k j (x) f (x)| 0}| r > 0, for some r > 0and for all j . By hypothesis there is a subsequence f kj m f a.e. as m . Nowuse Egorov to get a contradiction.

Problem 4.10 Let f : E R be a measurable function. Prove that if B R is aBorel set then f 1(B) is measurable.Hint: consider A = {A R : f 1(A) is measurable} and show that A is a-algebra that contains the open sets of R .

Problem 4.11 If f : E R is a measurable function and g : R R is Borelmeasurable, then g f is measurable.

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5 INTEGRATION

5 IntegrationProblem 5.1 Let f : [a, b] R be integrable and nonnegative. Prove that

b

af (x) cos x dx

2

+ b

af (x) sin x dx

2

b

af (x) dx

2

.

Hint: write f (x) = f (x) f (x)cos x and use Schwartz on the left hand side.Problem 5.2 Show that

limn

[0,1]n

x21 + + x2nx1 +

+ xn

dx = 2/ 3.

Hint: see Makarov, prob. 6.21, p.107.

Problem 5.3 Let f 1 : [0, M ] R be a bounded function. Denef n +1 (x) =

x

0f n (t) dt, n = 1 , 2, .

Prove that the series n =2 f n (x) is uniformly convergent on [0, M ] and the sum is

a continuous function on [0, M ].

Problem 5.4 Let f : Rn

R be integrable. Prove that1. |{x R n : |f (x)| = }| = 0.2. For each > 0, |{x R n : |f (x)| }| < .3. For each > 0 there exists a compact set K such that R n \ K |f (x)|dx < .4. For each > 0 there exist M R and A R n measurable such that |f (x)| M in A and R n \ A |f (x)|dx < .5. For each > 0 there exist > 0 such that if A Rn is measurable and suchthat

|A

| < then

A |

f (x)

|dx < .

Problem 5.5 Let f : E R be measurable with |E | < .Then f is integrable if and only if m |{x R n : |f (x)| m}| < .Hint: use Abels summation by parts formula,N

k=1

ak bk = AN bN +N 1

k=1

Ak (bk bk 1),

where Ak = k j =1 a j .

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5 INTEGRATION

Problem 5.6 Let f , f k : R n R be integrable functions such that f k f a.e.Then

R n |f k(x) f (x)|dx 0 if and only if R n |f k(x)|dx

R n |f (x)|dx.Problem 5.7 Let P [0, 1] be a perfect nowhere dense set, i.e., P

= , withpositive measure. Show that P (x) is not Riemann integrable on [0 , 1] but it isLebesgue integrable on [0, 1].

Problem 5.8 Let f : R n R be integrable and > 0. Prove that1. There exists g : R n R simple and integrable such that

R n |f (x)g(x)|dx


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5 INTEGRATION

7. f n (x) = 1 + x

n

nsin(

x

n

) in the interval (0 ,

).

8. f n (x) = 1 + nx(1 + x)n

Problem 5.17 Prove the following identities.

1. limn

n1/ 1

0x1/ (1 x)n

dxx

=

0e u

du if > 0.

2. limn

n

01

xn

nx( 1) dx =

0e x x( 1) dx if > 0.

3. limn

n e nx 2 dx =

( lim

n n e nx 2 ) dx if > 0.

4. 1

0

x1/ 3

1 x log

1x

dx = 9

n =1

1(3n + 4) 2

.

5.

0

sin tet x

dt =

n =1

xn 1

n2 + 1 for 1 < x < 1.

6. 1

0log x sin x dx =

n =1(1)

n

(2n)(2n)!.

Problem 5.18 If f is Riemann integrable on [a, b] and f (x) = 0 for x [a, b]Q ,then

ba f (x) dx = 0.

Problem 5.19 Show that

f (x) =1 if x = 1/n0 otherwise

is Riemann integrable on [0 , 1]. Find the value of 1

0 f (x) dx.

Problem 5.20 Let f : [0, 1] R be Riemann integrable. Prove that

limn

1n

n

k=1

f (k/n ) = 1

0f (x) dx.

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5 INTEGRATION

Problem 5.21 Let f : [0, 1] R be continuous, and f (x) = 0 for each x (0, 1).Suppose that

f (x)2 = 2 x

0f (t) dt

for each x [0, 1]. Prove that f (x) x.

Problem 5.22 Let f be a measurable function on [0, 1] and let

A = {x [0, 1] : f (x) Z}.Prove that A is measurable and

1

0[cos( f (x))]2n dx |A|,

as n .

Problem 5.23 Find all the values of p and q such that the integral

x 2 + y2 1 1x2 p + y2q dxdyconverges.

Problem 5.24 Let f 1, , f k be continuous real valued functions on the interval[a, b]. Show that the set {f 1, , f k} is linearly dependent on [a, b] if and only if thek k matrix with entriesf i , f j =

b

af i(x) f j (x) dx

has determinant zero.

Problem 5.25 Let f : [0, + ) R be continuously differentiable with compactsupport in [0, +

); and 0 < a < b 1}| = 0.

Problem 5.33 Prove that limn

sin x1 + n x 2

dx = 0.

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5 INTEGRATION

Problem 5.34 (Variant of Fatou) Let gn L1(E ) and gn g in L1(E ). Sup-pose f n are measurable functions on E and f n gn a.e. for each n. Prove that

lim supn E f n E limsupn f n .

Problem 5.35 (Variant of Lebesgue) Let gn L1(E ) and gn g in L1(E ).Suppose f n are measurable functions on E such that f n f a.e. or f n f inmeasure, and |f n | gn a.e. for each n. Prove that

E |f n f | 0,

as n , and consequently limn E f n = E f.Problem 5.36 Let |E | < . Prove that f n f in measure if and only if

E {|f n f | 1} 0as n .

Problem 5.37 Let f L1(0, 1) and suppose that lim x1 f (x) = A < . Showthatlim

n n

1

0xn f (x) dx = A.

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6 SOLUTIONS CONTINUITY

6 Solutions Continuity1.1 Let y < x with y irrational. Write

f (x) f (y) ={n :y r n 0 there exists N such that n N 1/ 2n < . Let = min {|y r k| : 1 k N }. If x y < and n is such that y rn < x , then n > N. Because if n N and y rn < x , then r n y < x y < which is impossible by denition of .If y Q , then y = rm for some m and so f (x)f (y) = {n :y r n y .

We have

1

0f (x) dx =

1

0

n =1

A(x)(n) 12n

dx =

n =1

12n

1

0 A(x)(n) dx

=

n =1

12n

1

0 (r n ,+ )(x) dx =

n =1

12n |[0, 1](r n , + )|.

1.2 Write

sin x + 4 n2 2

sin

y + 4 n2 2 = cos

+ 4 n2 2

1

2 + 4 n2

2

(x

y),

with some between x + 4 n2 2 and y + 4 n2 2. To show the pointwise convergencepick y = 0. For (3), |f n (z )| = 1 when z + 4 n2 2 = ( k + 1 / 2).

1.3 The class is bounded in C [a, b] and equicontinuous.

1.4 If |f (x)| 1 on [a, b], then T (f ) is bounded. Show that T (f ) are equicon-tinuous writing

T (f )(x) T (f )(z ) = b

a(x2 + z 2)f (y) dy +

b

a(e x

2

e z2

) ey f (y) dy.

1.5 The continuity follows from the uniform convergence of the series. Given > 0 there exists N such that k= N +1 2 k < . Write

f n (t) =N

k=1

2 kgn (t r k) +

k= N +1

2 kgn (t r k) = An (t) + Bn (t).

We have supp gn ( rk) [t 1/ 2n, t ]. Let E = {r 1, r 2, , rN }. Suppose rstthat t / E and let = dist( t, E ). If 1/ 2n < , then rk [t 1/ 2n, t ] for all

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1 k N and so An (t) = 0 for n > 1/ 2. If on the other hand t E , thenlim supn An (t)

N k=1 2

k

lim supn gn (t r k) = 0 .Let (a, b) be an interval. There exists r m (a, b) and so r m + 1 /n (a, b) for alln sufficiently large. Therefore sup x (a,b) f n (x) f n (r m + 1 /n ) 2 m gn (1/n ) = 2 mfor all n sufficiently large.1.6 f n (x) is a Riemann sum and converges to

x+1x f (t) dt as n . Let (a, b)be a nite interval. Write

f n (x) x+1

xf (t) dt =

n 1

k=0 x+( k+1) /n

x+ k/n(f (x + k/n ) f (t)) dt.

Since f is uniformly continuous on [a, b + 1] we have that

|f (x + k/n )

f (t)

| < for

all x [a, b], x + k/n t x + ( k + 1) /n , 0 k n 1, and for all n sufficientlylarge. Thereforesup

x [a,b]f n (x)

x+1

xf (t) dt <

for all n sufficiently large.

1.7 (a) (b) f (x) = 1 /x ; (b) (a) g(x) = 1, f (x) = 1 for x Q and f (x) = 0for x R \ Q .1.8 By the absolute continuity f (x) f (a) =

xa f (t) dt and g(x) g(a) =

x

a g (t) dt, for all a x b, so f (x) g(x) = f (a) g(a) for all a x b.1.9 f (x)2 f (y)2 = ( f (x) + f (y))( f (x) f (y)) and f is bounded.1.10 Let h > 0. We have |g(x + h) g(x)| |g(x + h) g(0)|+ |g(x) g(0)| (x + h)a + xa 2(x + h)a . Also |g (x)| 2a/x for x = 0. By the mean value theorem,

|g(x + h) g(x)| C a min{(x + h)a ,h/x }. Next either xa+1 h or xa+1 < h , andthen conclude g C ([0, 1]). To show the negative statement notice that g(x) = 0at x = = ( n ) 1/a with n = 1, 2, . Let > 0 small, and = (( n + )) 1/a .Suppose by contradiction that |g( ) g( )| K | | . We have |g (x)| =ax |xa sin x a cosx a |

ax

(|cosx a | xa |sin x a |) ax

(|cosx a | xa ). From themean value theorem |g( )g( )| = |g( )| | |, with (( n +1) ) 1/a (n ) 1/a .Choosing > 0 small, we get that |g ( )| C n

1/afor all n large. Also notice that

| | n (1+ a)/a for n large. Combining the orders of magnitude we obtain acontradiction unless a/ (1 + a).1.11 Let {f n}be a Cauchy sequence. Given x (a, b) take the partition {a,x,b}.We have

|f n (x)f m (x)| |(f n f m )(x)(f n f m )(a))|+ |(f n f m )(a)| f n f m BV 0,as n, m . Thus, for each x [a, b], {f n (x)} is a Cauchy sequence. Let f (x) =limn f n (x). Let = {x0, , xN } be a partition of [a, b]. Given > 0 there exists

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n0 such thatN

i=1|(f n f m )(xi) (f n f m )(xi 1))| <

for each n, m n0. Letting n yieldsN

i=1|(f f m )(xi) (f f m )(xi 1))| <

for each m n0. Hence f f m BV < for m n0.Let f n be absolutely continuous such that f n f BV 0. Given > 0there exists N such that

ki=1

|(f n

f )(xi)

(f n

f )(xi 1)

| < for each n

N

and any partition = {x0, , xk}. Since f N is absolutely continuous, thereexists > 0 such that if ( a1, b1), , (ar , br ) is any collection of disjoint inter-vals of [a, b] with ri=1 (bi ai) < , then

ri=1 |f N (bi) f N (a i)| < . I f =

{a, a 1, b1, a2, b2, , ar , br , b}, thenr

i=1|f (bi) f (a i)|

k

i=1|(f f N )(bi) (f f N )(a i)|+

k

i=1|f N (bi) f N (a i)|

f f N BV + < 2 .

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7 SOLUTIONS SEMICONTINUITY

7 Solutions Semicontinuity

2.6 Suppose y Q , y = rm . Then f (x) 12m

+ f (y) for all x > y , so

inf y 0. We now estimate inf y

real analysis problems - cristian e. gutierrez

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