INSTRUCTOR: CEZAR LUPU. Problem 1. a) Let f(x) be a continuous function on [1, 2]. Prove that. nx 2 lim

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1 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES INTEGRAL OF A SINGLE VARIABLE, INTEGRAL CALCULUS, FOURIER SERIES AND SPECIAL FUNCTIONS INSTRUCTOR: CEZAR LUPU Problem. Let f( be cotiuous fuctio o [, ]. Prove tht d =. + 4 b Let f( be cotiuous fuctio o [, ]. Prove tht d =. + 4 c Let f( be cotiuous fuctio o [, ]. Prove tht there eists β > such tht 3/ d = β f(. + 4 Uiversity of Pittsburgh Preiry Em, 6 Problem. Let f be C fuctio o [, ] such tht f( = d f for [, ]. Prove tht f(d 3. Uiversity of Missouri-Columbi Qulifyig Em, 995 Problem 3. Let f : [, ] R be cotiuous fuctio o [, ]. Prove tht f(t cos( t + 4 π dt = f(tdt. Problem 4. For ech N, let φ ( =. Suppose tht f( is cotiuous fuctio o [, ] which stisfies f(φ (d = f(φ m (d, for ll, m N. Prove tht f must be costt fuctio.

2 INSTRUCTOR: CEZAR LUPU Problem 5. Let f be cotiuous fuctio o [, ]. Prove tht tf( d = πf(. t + + t Ohio Stte Uiversity Qulifyig Em, 8 Problem 6. ( Let f C ([, ] such tht f( =. Prove tht f( ( / (f (. (b Let f C ([, ] such tht f( = f ( =. Prove tht f( ( / (f (. (c Let f C ([, ] such tht f( = f ( = f ( =. Prove tht f( ( / (f (. 3 Problem 7. Let f : [, ] R be cotiuously differetible, with f( =. Prove tht f where f = sup{ f(t : t }. (f ( d, Problem 8. ( Prove tht ( ( + e d = = (b Use prt ( to show tht = ( +. = ( + = log. Uiversity of Pittsburgh Preiry Em, 4 Problem 9. Prove tht, if f : [, ] R is cotiuous d f(d =

3 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES for ech iteger, the f o [, ]. Uiversity of Pittsburgh Preiry Em, 5 Problem. Suppose tht φ : [, b] R is differetible d f : (φ([, b] R is cotiuous. Prove the Leibiz rule: [ ] d φ( f(tdt = f(φ( φ (. d φ( Uiversity of Pittsburgh Preiry Em, 6 Problem. ( Prove tht if f : [, b] R is cotiuous, the ( log ep(f(d = sup f(. [,] (b Let f C([, ], R. Prove tht f( d = sup f(. [,b] Uiversity of Pittsburgh Preiry Em, 7 Problem. Let f : [, ] R be cotiuous fuctio. Show tht: ( (b (c (d f(d =. f( d = f(. f(d = f(. f( d = f(d. Problem 3. ( Let f : [, ] R be cotiuously differetible o [, ] d stisfy f( =. Show tht f( d 4 (f ( d. Uiversity of Pittsburgh Preiry Em, (b Let f, g C ([, b] such tht g(b =. Prove tht f (d = g (d = d f( =

4 4 INSTRUCTOR: CEZAR LUPU ( / ( / f(g(d (f ( + (g (. (c Let f d f be cotiuous o [, d f( = for. Show tht f (d f (d (f ( d. Berkeley Preiry Em, 995 Problem 4. [Riesz] Give u C([, ], defie f : [, ] R by f(tdt, (, ] f( = u(, = Prove tht ( / ( / f (d u (d. Uiversity of Pittsburgh Preiry Em, 3 Problem 5. Let f, g be two Riem itegrble fuctios o [, ] d h( = m{f(; g(} for [, ]. (i Prove tht h is Riem itegrble o [, ]. (ii Suppose tht {f } d {g } re two sequeces of Riem itegrble fuctios o [, ] such tht f ( f( d = Let h {f, g } for [, ] d N. Prove tht h ( h( d =. f ( f( d =. Uiversity of Pittsburgh Preiry Em, 3 Problem 6. ( Let f : [, ] [, be icresig fuctio. Show tht f k= ( k f(d f( f(. (b Let f : [, ] R be cotiuous fuctio tht is lso differetible o (, d the derivtive f is bouded o (,. Set M = sup f(t. Prove tht for ech <t< N oe hs

5 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES f j= ( j f(tdt M. Uiversity of Missouri-Columbi Qulifyig Em, 4 Problem 7. Let g : (, (, be cotiuously differetible fuctio with the property tht there eists two costts C > d M > such tht g( C d g ( M for ll >. Prove tht coverges. si g( d Uiversity of Missouri-Columbi Qulifyig Em, Problem 8. Let f : [, (, be cotiuous d periodic with period fuctio (i.e. f( + = f(, for ech, d defie Show tht F ( = f(tdt. t t t f( d = F (. Uiversity of Missouri-Columbi Qulifyig Em, 8 Problem 9. Suppose tht f : [, R is cotiuous o [,, differetible o (,, f( =, f (, for >. Show tht for, ( f(tdt f 3 (tdt. Uiversity of Missouri-Columbi Qulifyig Em, 7 Problem. ( Show tht if f : [, R is uiformly cotiuous d T f(tdt T eists d is fiite, the f( =. (b Let f : [, [, be mootoiclly decresig fuctio such tht Show tht f( =. f(d <.

6 6 INSTRUCTOR: CEZAR LUPU Uiversity of Missouri-Columbi Qulifyig Em, 5 Problem. Let f C ([, ] such tht f( = f( =. Show tht ( (f ( d f(d. Problem. ( Let (g be sequece of Riem itegrble fuctios from [, ] ito R such tht g ( for ll,. Defie G ( = g (tdt. Prove tht sequece of (G coverges uiformly. (b Let (f be sequece of rel vlued C fuctios o [, ] such tht, for ll, f (,, f (d =. Prove tht the sequece (f hs subsequece tht coverges uiformly i [, ]. (c Let (f be sequece of rel-vlued cotiuous fuctios defied o [, ] such tht f (y dy 3, for ll N. Defie g : [, ] R by g ( = + yf (ydy. Prove tht (g = cotis subsequece tht coverges uiformly. Problem 3. Evlute the followig itegrls: (i (ii (iii (iv (v (vi log + d. log( + d. rct d. log( d. log log( d. d. Uiversity of Licol-Nebrsk Qulifyig Em, 9

7 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES (vii (viii (i log + d. e d. e log d. Problem 4. Let f : [, ] R be cotiuous. Suppose tht f(g (d = for ech cotiuously differetible fuctio g : [, ] R stisfyig g( = = g(. Prove tht f must be the costt fuctio. Ohio Stte Uiversity Qulifyig Em, 5 Problem 5. Let the fuctio ϕ be cotiuous o [, ] with ϕ(d =, ϕ(d =. Prove tht there eists [, ] such tht ϕ( 4. Ohio Stte Uiversity Qulifyig Em, 4 Problem 6. Let f C ([, b]. Show tht there eist c, d (, b such tht d ( + b f(d = (b f + (b 3 f (c, 4 f( + f(b f(d = (b (b 3 f (d. Problem 7. Let f be cotiuously differetible fuctio o the itervl [, ] ito R. Suppose tht f(/ =. Show tht f( d f ( d. Ohio Stte Uiversity Qulifyig Em, Problem 8. Let f be rel-vlued cotiuous fuctio o [, ]. Prove tht d ( ep f(d ep(f(d,

8 8 INSTRUCTOR: CEZAR LUPU ( log f(d log f(d. Problem 9. [Frulli] Let f be rel vlued fuctio such tht f C([,, f( d the improper Riem itegrl coverges. Prove tht for ll > d b > we hve: f( f(b d = f( log b ɛ + ɛ. Applictios. Evlute the followig itegrls: (i (ii e e b d. rct(π rct d. Problem 3. ( Prove tht (b Prove tht π si + d = 4... ( ( +. rcsi = = c Deduce the Euler celebrted series, 3... ( 4... ( = = π 6. +, <. + Americ Mthemticl Mothly, 988 Problem 3. Let f : [, π] R be cotiuous fuctio such tht π f( si d =, for ll itegers. Is f is ideticlly zero? Problem 3. Let (φ = be sequece of oegtive Riem itegrble fuctios o [, ] which stisfy: (i φ (tdt = for ech.

9 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES (ii for every δ >, φ uiformly o [, δ] [δ, ]. ( Show tht if f : [, ] R is Riem itegrble d cotiuous t =, the (b Show tht φ (tf(tdt = f(. Problem 33. Evlute the it: e ( d = 4 3. Uiversity of New Meico Qulifyig Em, k= ( + k /k Problem 34. Let f : [, ] R be cotiuous fuctio such tht for ll [, ]. Show tht f(tdt, 3 f (tdt 3. Problem 35. Let [, ]. Show tht there is o cotiuous fuctio f : [, ] (, such tht f =, f =, f =. Problem 36. Let M = C([, ]. Defie d(, o M M by d(f, g = (f( g( d, = for f, g M. Prove tht (M, d is metric spce. Problem 37. Prove tht iff : [, ] R is cotiuous fuctio such tht

10 INSTRUCTOR: CEZAR LUPU e f(d = for ll =,,,..., the f( = for ll. Does the sme coclusio still hold true if f is odecresig? Problem 38. (i [Hermite-Hdmrd] Let f : [, b] R be cove fuctio. Show tht ( + b (b f (ii Show tht f(d (b f( + f(b. d k+ k f(d log k + log(k +, k, k+/ k / log d log k, k. (iii Cosider the sequece ( defied by = f(d log... log( log,. Show tht is icresig d log 5 4. (iv Prove tht 4 ( e 5 e ( e! e,. (v [Stirlig] Show tht! ( e π =. Problem 39. [Fejer] If f, g re cotiuous fuctios o R of period, the f(g(d = f(d g(d. Problem 4. ( [Poly] Show tht for f C([, b] C ([, b], f( = f(b =, f( d (b 4 sup f (. [,b] (b Show tht for f C([, b] C ([, b], f( = f(b =,

11 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES f( d (b 3 sup f (. [,b] ( + b Moreover, if f =, the the costt c be replced by 4. Problem 4. Let > d f( be cotiuously differetible o [, ]. Show tht f( f( d + f (d. Ohio Stte Uiversity Qulifyig Em, 99 Problem 4. [Wirtiger] Let f be twice differetible rel vlued fuctio o [, π], with π f(d = = f( = f(π =. Show tht π f ( k= π (f ( d. Problem 43. ( Let f : [, ] R be differetible, with f itegrble o [, ]. Show tht ( ( k f( f( f(d f =. Applictio. Evlute ( log k=. + k (b Let f : [, ] R be fuctio of clss C o [, ]. Show tht ( f(d ( k f = f ( f ( 4 k= Problem 44. Does there eist cotiuous rel-vlued fuctio f(,, such tht d for ll =,, 3, 4,...? f(d = f(d =,

12 INSTRUCTOR: CEZAR LUPU Problem 45. Evlute the followig: ( (b ( + e + e d. + ( log + ( α d, α (, ]. (c (d π π ( si. 4 ( = cos. + = Uiversity of Pittsburgh Preiry Em, (e (f ( ( s e /s ds, (,. ( d. + 4 = = Uiversity of Pittsburgh Preiry Em, 4 Problem 46. Let f be cotiuous rel-vlued fuctio o [, such tht ( f( + f(tdt eists d is fiite. Prove tht f( =. Problem 47. ( [Rogers-Holder] Let C([, ] deote the set of cotiuous relvlued fuctios o [, ]. Let p, q >, with p + q the ( f(g( d =. If f d g re i C([, ], /p ( /q f p (d g (d q. (b [Mikovski] Let p (,. For ech fuctio f C([, ], let f p := ( /p f(d. Prove tht f + g p f p + g p. Problem 48. [Hrdy] Show tht if f : [, R + is itegrble d p >, the oe hs

13 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (RIEMANN AND RIEMANN-STIELTJES ( f(tdt p d ( p p f p (d. p Problem 49. Let (f be sequece of cotiuous fuctios defied from [, ] ito R such tht (f ( f m (d,, m. Let K : [, ] [, ]tor be cotiuous. Defie g : [, ] R by g ( = K(, yf (ydy. Prove tht the sequece g coverges uiformly. Problem 5. Let ϕ, ϕ,..., ϕ be oegtive cotiuous fuctios o [, ] such tht the it k ϕ (d eists for every k =,,.... Show tht the it f(ϕ (d eists for every cotiuous fuctio f o [, ]. Berkeley Preiry Em, 98 Problem 5. Suppose tht f : [, ] R hs cotiuous derivtive d tht f(d =. Show tht for every α (,, α f(d 8 sup f (. Putm Competitio (Problem A, 7 Problem 5. Let f : [, ] R be differetible fuctio with bouded derivtive. Show tht ( f (d Problem 53. ( Show tht f(d ( sup f (. e e t dt =.

14 4 INSTRUCTOR: CEZAR LUPU (b Prove tht Uiversity of Pittsburgh Preiry Emitio, 5 sup e e t dt =. Ohio Stte Qulifyig Em, Problem 54. Let f : [, R be bouded d cotiuous. Prove tht sup b b f(d sup f(. Ohio Stte Qulifyig Em, 3 Problem 55. Let f C([, ] d Prove tht f is odd fuctio. f(d = for ll itegers. Ohio Stte Qulifyig Em, 9 Problem 56. [v der Corput] Let φ C (, (recll tht this mes tht φ isifiitely differetible d φ is ideticlly i some eighborhood of d. Show tht for y turl umber N, there eists costt C N such tht e iλ φ( Cλ N, for ll λ >. Ohio Stte Qulifyig Em, 4 Problem 57. Suppose tht f : [, R is cotiuous. Prove tht if f(d eists (i short, for every α > d α + Problem 58. Show tht f(d coverges, the e α f(d = f(d. ( d = π. e α f(d coverges Ohio Stte Qulifyig Em,

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.