= s'[(1-t')-(1-t)]-s[(1-t')-(1-t)]
= (s'-s)(t-t') 5 0.
ii) If we set g(s,t) = -f(s,t), we get a function g satisfying
Definition 2.1.3(b), but not (a):
g(s',t')-g(s,t) = -f(s',t')-(-f(s,t))
= -[f(s',t')-f(s,t)]
5 0,
and similarly A ,g = -(A ,f) > 0. We could even create a
zz zz
nonnegative g (g = 1-f) with these properties.
As can be seen, these two definitions of increasing are not
nearly so closely related as the definitions of right continuity we
have given. Later, however, we shall give sufficient conditions for a
function f of two variables to have a "Jordan decomposition"
f f-f 2, where f and f2 are increasing in both senses of the word.
2.2 The Variation of a Function of Two Variables
In this section we define the variation Var z,z',(f) of a
2 2
function f: R2 + E on a rectangle [z,z'] (closed) in R and establish
some of its properties. Throughout this section, by "rectangle" we
shall mean a closed, bounded rectangle in R2 (but everything goes
equally well for such rectangles in R ), unless otherwise specified.
Definition 2.2.1. Let R = [(s,t),(s',t')] be a closed, bounded
rectangle in R2