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文章是讲一致收敛和微分的~我英语不好看不明白~请大家帮帮忙啊~~~~ Theorem 9.13. Assume that each term of is a real-valued function having a finite derivative at each point of an open interval
# ?% h& \' a: d. o7 b, h7 V . Assume that for at least one point2 T: Y9 E6 f C" Z) c
in
6 S% q: `6 y9 e/ X# b4 Y N8 c* j the sequence converges. Assume further that there exists a function g such that
! J; \5 g( D3 @4 I uniformly on
/ R# d; S8 u7 L) d . Then: / C0 M5 m. s6 q7 H' a0 m
a) There exists a function f such that
- ^6 B/ e1 w4 J uniformly on {& \1 l; V! S( i! f% N
. b) For each x in% g! Y4 g$ l! {( c
the derivative
% v$ y7 ?0 T( k0 { exists and equal
! q7 X$ b' g) ^ l% h/ m0 x . Proof. Assume that
1 O h1 N$ ~( Q2 z6 i# t and define a new sequence
4 q9 i8 @, g* n L2 d as follows:
4 x! U$ x& n1 z1 C# c( D
/ N$ ? Y- q: {; ~ (8)
|0 j0 _& W7 k/ X% p) x
The sequence
3 S( p0 J- `) [ so formed depends on the choice of c. Convergence of follows from the hypothesis, since
: N+ H& R# `* ?& B; [ . We will prove next that6 d0 S" ^9 L% f. W
converges uniformly on9 b5 e8 R/ L# N& ]2 m2 ^; P5 k5 G& q
. If , we have
7 G: E. c. c ?+ D* t9 I3 U9 f
,3 e' g6 u- ^: g, `
(9)
/ b. q3 \, N5 h6 I1 w5 ?; M+ N
where
& L* S: F' n% o$ q# t) z5 d . Now
# T* {! A7 N6 p& ` exists for each x in
+ A; X0 J% b3 E and has the value
: f( ^' q! j6 C . Applying the Mean-Value Theorem in (9), we get ,
& T' k. G$ M; y$ F2 ^ 1 o. V0 h% K1 \3 z5 i
(10) where6 m- R# S. }8 H1 f* U2 f
lies between x and c. Since
0 N: J$ w# w/ | converges uniformly on
2 H1 n" d6 b& N (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that2 W1 E. C' I& }
converges uniformly on3 v# y% v: ?* W8 \, w2 E
. Now we can show that5 [. ^& z L1 ~
converges uniformly on- y: [7 T+ \3 W5 L2 P
. Let us form the particular sequence
# X0 c' P+ i, p l& j9 _ corresponding to the special point6 u7 M' l2 b( Z- I3 D
for which+ ?8 n, N: J# p
is assumed to converge. Form (8) we can write
an equation which holds for every x in# Z4 _# w4 c l4 e
. Hence we have
This equation, with the help of the Cauthy condition, establishes the uniform convergence of on
# h7 y# X" U7 @3 I2 e1 z/ f- f . This proves (a). To prove (b), return to the sequence& @; d( H/ f+ I. \
defined by (8) for an arbitrary point c in4 x1 C$ G! ], ]8 e c
and let/ [$ G) z( U+ l! t Z
. The hypothesis that/ \. A/ p$ o* }+ n3 P% {5 X
exists means that . In other words, each
7 j2 E; Y( W" i# t- l# x% V is continuous at c. Since
3 s1 i; `1 ?( ^6 e uniformly on. d! m3 L# t* M6 n5 ?
, the limit function G is also continuous at c. This means that
% f8 |, ~, D: q4 C7 E4 v8 t% t (11) the existence of the limit being part of the conclusion. But for
3 H/ J1 m6 E* ^" f* X( i , we have
Hence, (11) states that the derivative
) O( x, l) h* O$ w ^ exists and equals
5 J/ [; @; q6 F . But
hence0 D3 _- M4 k4 W' `4 i* T; K% u
. Since c is an arbitrary point of
! N. P* }" J; I/ ]; q , this proves (b). When we reformulate Theorem 9.13 in terms of series, we obtain
9 N2 R- h( e1 Q+ h
Theorem 9.14. Assume that each
9 O* E5 P6 |# L& T; I' W is a real-valued function defined on
* h0 J' |* T1 l6 u) j such that the derivative* Y! P u* G, C# z( t: J/ I: \
exists for each x in7 ?! b$ d! H$ W K
. Assume that, for at least one point
. K2 v, v" R+ R( I in! }5 Y% E1 K: ^% Y3 X, g
, the series6 o. @- ~7 B. m, N" x; e2 Q
converges. Assume further that there exists a function g such that (uniformly on
' d! y8 E( F# i& U ). Then: a)
$ H# t; a. Q) j" c( ] There exists a function f such that: F# Z' O8 n8 d# n, J( e M* q
(uniformly on( R1 v: ~# u$ `7 ?* }. j) K. A' ?
). b)
4 r9 v* Z: Y' h2 T( ^9 ]/ B If , the derivative. _. k: H' Y' y% n
exists and equals
! K* Y$ ]: d E/ t- ]) H: U3 c . |