right-limits everywhere, there exists a number p > 0 common to all the
vertices of all the Jh such that
If(sh+,th+) f(sh Pth P)I
If(sh+,th+) f(sh p,t'+p)I < C
Wf(sh ,t+) f(sh+p,t^+p) < and
|f(s+,'th+) f(sh+P,th+p)| < i for all h.
n
If we denote Ih = [(s +p,th+p),(s'+p,t'+p)], then I = ) Ih is a
h h h h h h=1
h=1
closed, bounded rectangle in R2, and the family P = {Ih: h = 1...n}
forms a partition of I according to Definition 2.2.1. Also for each
h, we have
IJh (f+) A (f)
h h
= If(s+,t+) f(s+ 'th+) f(sh+t+) + f(s th)
(f(sh+p,th+p) f(sh+Pth+P) f(sh+pi,t+p) + f(sh+Pth+P))I
= lf(s ,'th+) f(sh+P,th+p)) (f(sh+,t +) f(s+Pth +))
(f(sh+,th+) f(sh+P,t +p)) + (f(h+,t h+) f(sh+Pth+P))h
+ If(sh t+) f(sh+Pt'+P)I + If(sh+'th+) f(sh Pth+P)
E E E
-- u+ + "