Конспект лекций по математическому анализу. Шерстнев А.Н. - 134 стр.

KOTOROE NUVNYMI SWOJSTWAMI OBLADAET.g tAK KAK f NEPRERYWNO DIFFE-
RENCIRUEMO, SU]ESTWUET U = BR() TAKOE, ^TO
(2)                       kf 0(x) , I k < 1=2 (x 2 U ):
nERAWENSTWO (2) OBESPE^IWAET, W ^ASTNOSTI, ^TO LINEJNOE OTOBRAVENIE
f 0(x) OBRATIMO PRI L@BOM x 2 U . |TO SLEDUET IZ OCENKI kf 0(x)(h)k 
khk , k(f 0(x) , I )hk  12 khk S U^ETOM 73.2.
    pOLOVIM V = f (U ). tAKIM OBRAZOM, f : U ! V | S@R_EKCIQ PO PO-
STROENI@. iTAK, NEOBHODIMO USTANOWITX, ^TO (A) f : U ! V | IN_EKCIQ,
(B) V OTKRYTO, (W) IMEET MESTO FORMULA (1).
    pROWERIM (A). pUSTX x; x + h 2 U PROIZWOLXNY. rASSMOTRIM WEKTOR-
FUNKCI@ F (t)  f (x + th) , th (0  t  1). tOGDA (TAK KAK x + th 2 U (0 
t  1)) IMEEM dF (t) = (f 0(x + th) , I )(h)dt: oTS@DA S U^ETOM (2)
                                                   Z1
   kf (x + h) , f (x) , hk = kF (1) , F (0)k = k 0 (f 0(x + th) , I )(h) dtk
                                 Z1
                              0 kf 0(x + th) , I kkhk dt  12 khk:
pO\TOMU
(3)                         kf (x + h) , f (x)k  12 khk;
TO ESTX f : U ! V | IN_EKCIQ.
      (B). pUSTX x 2 U I r > 0 TAKOWO, ^TO Br [x]  U . pOKAVEM, ^TO
B 14 r (f (x))  V (OTS@DA SLEDUET, RAZUMEETSQ, ^TO V OTKRYTO).
      iTAK, PUSTX WEKTOR y 2 B 41 r (f (x)) PROIZWOLEN. pOKAVEM, ^TO SU]EST-
WUET x 2 Br (x) TAKOJ, ^TO f (x) = y. |TO O^EWIDNO, ESLI y = f (x) (TOGDA
x = x). pUSTX y =   6 f (x). wWEDEM FUNKCI@
                        '(u) = ky , f (u)k2 (u 2 Br [x]):
w SILU 70.2 SU]ESTWUET TO^KA x 2 Br [x] TAKAQ, ^TO '(x) = u2inf   Br [x]
                                                                           '(u).
nA SAMOM DELE x 2 Br (x) (DEJSTWITELXNO, RAWENSTWO kx , xk = r WLE^ET
S U^ETOM (3)
         1 r  kf (x ) , f (x)k  '(x )1=2 + '(x)1=2 < '(x )1=2 + 1 r;
         2                                                     4
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