Advanced Calculus of a Single Variable by Unknown

Author:Unknown [Unknown]
Language: eng
Format: epub
Published: 2016-02-12T12:35:47+00:00

ˇ

ˇ

f . k/ xk

f . x/ dxˇˇ < "

k D1

a

and

ˇŽ

ˇ

ˇ n

X

b

ˇ

ˇ

ˇ < "

ˇ

g . k/ xk

g . x/ dxˇˇ

k D1

a

if P D f x g n

g n

k k D1 is a partition of Œ a; b such that jj P jj < ı and f k k D1 are corre-

sponding intermediate points. We can assume that the subintervals I 1; I 2; : : : ; IN are

among the subintervals J D

k

Πxk1; xk, k D 1; 2; : : : ; n. Let us set

n

X

‚…„ƒ

X

f .

D b

P

j C

j

k/ xk

f . k/ j Jk

f . k/ j Jk ;

k D1

n

X

‚…„ƒ

X

g .

D b

P

j C

j

k/ xk

g . k/ j Jk

g . k/ j Jk ;

k D1

where the first sum is over those subintervals other than I 1; I 2; : : : ; IN and the second

sum is over the subintervals I 1; I 2; : : : ; IN. Note that

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