lim
(s',t') (s,t)
(s',t')M(s,t)
f(s',t') = lim f(s',O) = f(s,0)
s'++s
(Recall that the definition of right continuity allows us to take
limits along vertical or horizontal paths as well, unlike left
limits.) Hence f(-,O) is right continuous, so the variation is right
continuous, i.e., Var ,f(-,O) = lim Var f(*,O). Similarly,
,stak = 0, we hav
taking s' = s = 0, we have
f(C,t) = f(s,t) -
lim
(s',t')(s,t)
(s',t')(s,t)
f(s',t') = llm f(0,t'),
t'++t
so f(0,-) is right continuous. The variation is then right
continuous, so Var ,t]f(0,-) = lim Var[,t ]f(0,-). Then each of
t't ,t'
the terms of Ifl is right continuous; hence Ifl is right continuous on
R2
R.
Conversely, assume Ifl is right continuous at each point
2
(s,t) E R Taking t = 0, and letting s'4+s along the path t = 0, we
have
Var os' f(-,O) Var[0,s]f(.,O) Ifl(s',0) Ifl(s,0).
In fact,