@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$u)Er, @) ?tt"(T)+ouu(nr))f:', (y) ft;ffi9f:r.B) Nu anobeuX0ei 6rr limr*ro ouu(r) : ouu(rs),6rou z6 e -R.
@EMA ao. (MON. 2)A) Na e€erqo0e( av 4 ouvctptqo4
elvul ouvelllq oe x¬o oqpeio tou ,R,B) 'Eotco
f : R -+ ft ouvel{e ouv&prrloq t6tote drore
f (2) +/ (5) < 7 < f (3) + f (4)"
Nu anoDelplo6v 6rl un&p1ouv { xcu. ( npctlpcrrtxo( apt0poi t6rolol ciote: {a(: 7xar / ( { ) + f (0 :7 .
@EMA 5o. (MON.2)A) Na ppe0e( 4 ftapciyoyoq torv ouvaptrioeov:
(o) /(") : r1p3(1/7 +t) xar p) f (*) : tig#fi'*J'
B) II6re pr,a ouvdpqo\ f : A I R -+ fi xstreircrr opor,6;ropcpu ouvepl<; Xt4ouv6Xetc va e{eroo0ei av or nopcxdlc,r ouvaptllo'et< f ,g,h eivou o;ror6pcpo ouveleiq:
( o ) / ( r ) = 1 2 , r e R , ( 9 ) g ( r ) : * , * € ( 0 , + o o ) , ( y ) h ( " ) : 5 o * 1 0 ,r € R .
f) 'Eorco f r R -+ Q ouvepi< ouvdprqorl. Na anober,X0el6rr, 4 f eivut ota0ep{.
KotI{ Enrtul[oc
r(*):{t',:EIrn
s Ano.'tncus. -_!l?Almrr\x$ Arvern.l f H T(l) 9g$d. +t \ (x).r xQ srvo-t 6wtx'1! Gro L ,, 0, uno{pxar
r . ;Fo)r o "( \ \ " ( tc- , ) l ( r )o { )o,( t )Yr ln l .N \ Sar&,r . r , nn(p(NF6. l , " t 6<o x=Q,on6(e
tlvor, (\ o.Q r b,-,\,6.|,",
Gr " G 5, 5),(f) A)"r$+.,S. G x X,r$r xe R^ E{ovl.,t X- Qoll < x-2o10.
€nak n$ e^-*- 6'.1n6iq dv\.0vto1 $ ( x.tou)< \ Cx^goto),
"fr,;lj*tll't ;'il: rs6 ' rtpon*oa 3'd'g 6€)'s"( ttr'
A)'AGrnq 8 6t\itu r*q Srho"* :: ^:r,[:t;I : 'n
(+1 gtu\rs, t{ <r.toSH ", ( \cr)n l)- 6'6ili-u '<$ v
\ o(ro\"$A" (C-t)o)- S't^r 6.,y\iv€^ (d\rn. eK'Co tr+).(5) tptu!4 r ,'?xo+t f.X-1^
- o Ae" 0*1"* (dn) ' o
[elrnr Ttginua,r g.8,5 6r\iS*ilt*(6) qrril l, €cn- A= l-e,-:] rx1 g'ts,{. r[l"tt A'B= lO,-6,-t'
enioTg s.^rCA). Sqp($) =-2 . 3 =-6 + s"lCt'B)= - +
(T) A \'rs;l ' B \ent Agottrtry T- \'1 6t\ 1f 't 3t5
(s) Ys,$hr, €G<*' i:fo,'1 *", 9=l r,t,1\' Tort
A-g= &,-tr^x,*3]. on.n€ s'rr(A) -s..r?CB)o !-l ='- o + o;iltj:?
€MA i .o
U oe Gt,r)
euA to,
MA 3o
€eprA
€'vtA 5o
{w
o ru r" irit" t*r '0h6rn,rq re 6r\i$'( e"B!' '
t ;Ut"\ li' ii* Jlf Y\l'*hv'('(6/ o *i r'^-'3t f l .ott 3t4
li n;;* r 6e\ ' [x l+4 '
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