Θέματα Εξετάσεων Πραγματική Ανάλυση Ι Ιουνιος 2015(1)

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@EMATA EEETA>EON flpocapatrxll AvciLuorl I Io0vroq 2O15 @EMA l-o. (MON. 2) Na lapaxqpro0o6v (pe ocrtroL6ttlot1) ot rcxp(- xcirco npotdoerqcoq A).10e(<(A) ri ileubeiq (V): (1) Av / ouvripr4o4 ouvepie oro fa, b] xot un&p1er rs € (a,b) r,rote .f ("0) : 0, r6re xan' evcilx4 0u roX6eu /(a) ' /(b) < 0. (2) H ouvdpt4orl /(r) : lrl txovonorei tr,( npouno0doerq tou Oecopripmoq Rolle oto [-5,5]. (3) Av f ehtat pra ouv&pr1o4 yvqo(coq cr0{ouoaoto .R, t6te r,o1Oet f (r - 2011) < f (r - 2010), yrctx<i0er e R. (a) Av 4 axo),ourlioc npuyparuxclv epr0pcirv (lo,l)f,r ouyx).ivel, r6te xm 11 q,xo).ou0(ct (a ") -?= t ouyx),[ve l (5) 'Eotco q crxotrourXu (o.)ff:t VE an: fi. T arr limsup,.**(o,) : *. (6)'EorcoA,B,A. B qpaypdva unooOvo).cx'cou R, 6xov A. B : {a.b : a e A, b e B). T6re sup(A) 'sup(A) : sup(,A'B). (7) tsotco A C R. Av / : A -+ R opor.6popga ouvelll< ouvriprqo'4,t6te 4 / elvcr.L xcr,r ouve14C. (8) 'Eot<,1 A,B,A - B cppcryp6vq, unoo0vo),c rou -8, 6trov A- B - {a-b: a e A, b e Bj. T6te sup(.A) - sup(B) : sup(A - B). @EMA 2o. (MON. 2) Na onobetXrlei 6tl xd0s qxo).ou0(q, Cauchy elvar gpayp6v4. Xtrl o'uv61era. vcr anobelX0ei 6tr ptu exo).ou0ia npalpcttrxtirv uplr9prcilv (or)ff i ouyx),iveto''evo npeypcrtlx6upr0p6r6re xql pt6vo r6re 6rev eivcu, Cauchy. @EMA 3o. (MON.2) A) Nc uno).oyro0o0v(crvun&p1ouv)ra 6ple tolv q,xo).ou0tciv: 1e; (lt$u)Er, @) ?tt"(T)+ouu(nr))f:', (y) ft;ffi9f:r. B) Nu anobeuX0ei 6rr limr*ro ouu(r) : ouu(rs),6rou z6 e -R. @EMA ao. (MON. 2) A) Na e€erqo0e( av 4 ouvctptqo4 elvul ouvelllq oe x&noto oqpeio tou ,R, B) 'Eotco f : R -+ ft ouvel{e ouv&prrloqt6tote drore f (2) +/(5) < 7 < f (3) + f (4)" Nu anoDelplo6v 6rl un&p1ouv { xcu. ( npctlpcrrtxo( apt0poi t6rolol ciote: {a(: 7 xar /({) + f (0 :7. @EMA 5o. (MON.2) A) Na ppe0e( 4 ftapciyoyoqtorv ouvaptrioeov: (o) /(") : r1p3(1/7 +t) xar p) f (*) : tig#fi'*J' B) II6re pr,a ouvdpqo\ f : A I R -+ fi xstreircrr opor,6;ropcpu ouvepl<;Xt4 ouv6Xetc va e{eroo0ei av or nopcxdlc,r ouvaptllo'et< f ,g,h eivou o;ror6pcpo ouveleiq: (o) /(r) =12,reR, (9)g(r) :*,* € (0,+oo), (y) h(") :5o*10, r€R. f) 'Eorco f r R -+ Q ouvepi< ouvdprqorl. Na anober,X0el6rr, 4 f eivut ota0ep{. KotI{ Enrtul[oc r(*):{t',:EIrn

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Transcript of Θέματα Εξετάσεων Πραγματική Ανάλυση Ι Ιουνιος 2015(1)

Page 1: Θέματα Εξετάσεων Πραγματική Ανάλυση Ι Ιουνιος 2015(1)

@EMATA EEETA>EONflpocapatrxll AvciLuorl I

Io0vroq 2O15

@EMA l-o. (MON. 2) Na lapaxqpro0o6v (pe ocrtroL6ttlot1) ot rcxp(-xcirco npotdoerq coq A).10e(< (A) ri ileubeiq (V):

(1) Av / ouvripr4o4 ouvepie oro fa, b] xot un&p1er rs € (a,b) r,rote .f ("0) : 0,r6re xan' evcilx4 0u roX6eu /(a) ' /(b) < 0.

(2) H ouvdpt4orl /(r) : lrl txovonorei tr,( npouno0doerq tou OecopripmoqRolle oto [-5,5].

(3) Av f ehtat pra ouv&pr1o4 yvqo(coq cr0{ouoa oto .R, t6te r,o1Oet

f (r - 2011) < f (r - 2010), yrct x<i0e r e R.(a) Av 4 axo),ourlioc npuyparuxclv epr0pcirv (lo,l)f,r ouyx).ivel, r6te xm 11

q,xo).ou0(ct (a ") -?= t ouyx),[ve l

(5) 'Eotco q crxotrourXu (o.)ff:t VE an: fi. T arr limsup,.**(o,) : *.

(6) 'Eorco A,B,A. B qpaypdva unooOvo).cx 'cou R, 6xov A. B : {a.b : a eA, b e B) . T6re sup(A) 'sup(A) : sup( ,A 'B) .

(7) tsotco A C R. Av / : A -+ R opor.6popga ouvelll< ouvriprqo'4, t6te 4 /elvcr.L xcr,r ouve14C.

(8) 'Eot<,1 A,B,A - B cppcryp6vq, unoo0vo),c rou -8, 6trov A- B - {a-b:

a e A, b e Bj. T6te sup(.A) - sup(B) : sup(A - B).

@EMA 2o. (MON. 2) Na onobetXrlei 6tl xd0s qxo).ou0(q, Cauchy elvargpayp6v4. Xtrl o'uv61era. vcr anobelX0ei 6tr ptu exo).ou0ia npalpcttrxtirv uplr9prcilv(or)ff i ouyx),ivet o''evo npeypcrtlx6 upr0p6 r6re xql pt6vo r6re 6rev eivcu, Cauchy.

@EMA 3o . (MON.2 )A) Nc uno).oyro0o0v (crv un&p1ouv) ra 6ple tolv q,xo).ou0tciv:

1e; (lt$u)Er, @) ?tt"(T)+ouu(nr))f:', (y) ft;ffi9f:r.B) Nu anobeuX0ei 6rr limr*ro ouu(r) : ouu(rs),6rou z6 e -R.

@EMA ao. (MON. 2)A) Na e€erqo0e( av 4 ouvctptqo4

elvul ouvelllq oe x&noto oqpeio tou ,R,B) 'Eotco

f : R -+ ft ouvel{e ouv&prrloq t6tote drore

f (2) +/ (5) < 7 < f (3) + f (4)"

Nu anoDelplo6v 6rl un&p1ouv { xcu. ( npctlpcrrtxo( apt0poi t6rolol ciote: {a(: 7xar / ( { ) + f (0 :7 .

@EMA 5o. (MON.2)A) Na ppe0e( 4 ftapciyoyoq torv ouvaptrioeov:

(o) /(") : r1p3(1/7 +t) xar p) f (*) : tig#fi'*J'

B) II6re pr,a ouvdpqo\ f : A I R -+ fi xstreircrr opor,6;ropcpu ouvepl<; Xt4ouv6Xetc va e{eroo0ei av or nopcxdlc,r ouvaptllo'et< f ,g,h eivou o;ror6pcpo ouveleiq:

( o ) / ( r ) = 1 2 , r e R , ( 9 ) g ( r ) : * , * € ( 0 , + o o ) , ( y ) h ( " ) : 5 o * 1 0 ,r € R .

f) 'Eorco f r R -+ Q ouvepi< ouvdprqorl. Na anober,X0el6rr, 4 f eivut ota0ep{.

KotI{ Enrtul[oc

r(*):{t',:EIrn

Page 2: Θέματα Εξετάσεων Πραγματική Ανάλυση Ι Ιουνιος 2015(1)

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\ o(ro\"$A" (C-t)o)- S't^r 6.,y\iv€^ (d\rn. eK'Co tr+).(5) tptu!4 r ,'?xo+t f.X-1^

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[elrnr Ttginua,r g.8,5 6r\iS*ilt*(6) qrril l, €cn- A= l-e,-:] rx1 g'ts,{. r[l"tt A'B= lO,-6,-t'

enioTg s.^rCA). Sqp($) =-2 . 3 =-6 + s"lCt'B)= - +

(T) A \'rs;l ' B \ent Agottrtry T- \'1 6t\ 1f 't 3t5

(s) Ys,$hr, €G<*' i:fo,'1 *", 9=l r,t,1\' Tort

A-g= &,-tr^x,*3]. on.n€ s'rr(A) -s..r?CB)o !-l ='- o + o;iltj:?

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