= Var (f) ( Van (f)
ar[(o.t),(s',t')] (ot),(st')]
= Var (f) 0.
= a [(s ,t ) ,(s '. t .)] f O .
This definition allows us to recover many results analogous to
those of functions of one variable, as we show in the next few
theorems.
Theorem 2.3.2. Let f: R2 E have finite variation Ifl. Then f is
right continuous if and only if Ifl is right continuous.
Proof. Assume, first, that f is right continuous. We write, for
(s,t) (0,0),
Ifl(s,t)
= If(0,0)l + Var o f(,O) + Vartf(O,-) + Var [(O)(st)](f).
The first term is constant; to show that Ifl(s,t) = lim IfI(s',t'), it
s'+s
t'+t
suffices to show that
i) lim Var f(-,O) = Var ,s]f(,O0)
s ++s ,s'] 10,s]
ii) lim Var [ ,t ]f(O,.) = Var ,t]f(0,.)
t' +t
iii) lim Var (O)( (f) = Var (f).
(s',t')++(s,t)]
We proved (iii) in Theorem 2.2.6 (taking z = (0,0), z' = (s,t),
u = (s',t')). As for the other two, f right continuous implies
f(',0), f(O,-) right continuous: in fact, taking t' = t = 0, we have