Integration 425
Also, by Theorem 6.38, for a given 'f/ > 0 there exists a 6 > 0 such that
for any partition P with IPI < 6 we have 'f/P < "I· Hence from (7.8-2)
and (7.8-3) it follows that
lzk - Zk-11 < (M + 'fJ)I<
for any partition with IPI < 6. This proves that the function z = g(((t))
is of bounded variation over [a, ,8], since for any two partitions P 1 and P 2
with P 1 C P 2 (i.e., P 2 a refinement of Pi) we have
Vf(h,P1):::; Vf(h,P2)so that variation of a function h over P 1 is not greater than the variation of
hover P 2 • In other words, it suffices to prove the boundedness of the sum
I: lzk - Zk-1 I over partitions with norm less than some positive number.
Thus the arc 'Y is also rectifiable.
From (7.8-1) we have
Zk - Zk-1 = g'((k-1)((k -(k-1) + €k((k - (k-1)
and
n n
L f(Zk-1)(zk - Zk-1) = L f(g((k-1))g'((k-i)((k - (k-1)
k=l k=l
n
- L €kf(zk-1)((k - (k-1) (7.8-4)
k=l
Since f is continuous along "f, for some Mi > 0 we have lf(z)I :::; Mi for
all z E 1*. Hence
I~ '>f(z,_,)((k -(•-di,; "pM, ~I(, -(•-d,; "pM,Lb')
This proves that the last sum in (7.8-4) tends to zero as IPI --+ O, for 'f/P --+ 0
as IPI --+ 0 Thus taking limits in (7.8-4) as !Pl --+ 0, we obtain
j f(z) dz= j f(g(())g'(() d( (7.8-5)
'Y 'Y'(b) Continuously differentiable arc. If 'Y' is of class C^1 it follows that
1: z = g(((t)) is also of class C^1 since z' = g'(((t))('(t) exists and is
continuous for a :::; t :::; ,8. Then by Definition 7.3 we have
J f(z) dz= 1: f(g(((t)))g'(((t))('(t) dt
'Y