Now, let J E 6, J = ((s,t),(s',t')] with s~~0.
There exists a point (p,r) E J such that for any point (h,k) with
(s,t) < (h,k) < (p,r) we have If(h+,k+) f(s+,t+)l < ,
If(s+,t'+) f(h+,t'+) < f(s'+,t+) f(s'+,k+) < as in
Figure 3-2. (We can do this for each by the above, and we use a
common n in choosing our (p,r).)
(s,t') (h,t') (s',t)
I I
II
I
I I (p,r)
S -----------(s',k)
(h,k)
(s, t~ ~ ~ T- s,t)
Figure 3-2 Approximation of a rectangle from within
Let, then, K = [(p,r),(s',t')] compact. Any rectangle J' from
6 such that K C J'C J must be of the form J' = ((h,k),(s',t')]
with (s,t) < (h,k) < (p,r) (see Figure 3-2). We have, then,
jo(J) o(J') IA(f+) J,,(f+)l
= f(s'+,t'+) f(s+,t'+) f(s'+,t+) + f(s+,t+)
~~