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v ? , v, v v . vvv . f(x)>0 [a, b] vv " " v . T(f, a, b) . v v vv :

[a, b] [t0, t1], [t1, t2], [t2, t3], [t3, t4] . v =t0<t1<t2<t3<t4=b .

T(f, a, b) - . [t0, t1] f max - M1, min - m1 . - vv

s=m1(t1-t0) + m2(t2-t1) + m3(t3-t2) + m4(t4-t3),

vv

S=M1(t1-t0) + M2(t2-t1) + M3(t3-t2) + M4(t4-t3)

. s, S T(f, a, b) - .

̺ [a, b] , v s T(f, a, b) S .

a<b . [, b] P , .

vv t0,...,ti v = t0 < t1 <...< ti-1 < ti = b . ( .)

f [a, b] P - [a, b] - v. mi [ti , ti-1] min, Mi [ti , ti-1] max L(f, P) ,
, U(f, P) ,
.

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. v, v [a, b] . ( v v.) , {0<x1 1/x, =0 0} [0, 1] v . , -->0 f(x)--> . , . , [a, b]. v. , (a, b). v v . ( , , ....). vv vv ( 㺺 ) vv .

. 1, 2 L(f,P1) U(f,P2). , , , .

v , v .

vv - , - . - - inf {A} . - - sup {A} .

inf, sup . ( v vvv .) ...

P, P' ,

L(f,P') sup {L(f,P)} inf {U(f,P)} U(f,P')

. sup {L(f,P)} = inf {U(f,P)} T(f, a, b) . (v ) v v :

f [a, b] . sup {L(f,P)} = inf {U(f,P)} f [a, b] f .

. ( sum v s - .) , v . vv. v. ( v !)

. xi-xi-1 " - " vvv i .

vv ,

. ( ? :) ) "dx" dx - .

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