(Note: We shall use sometimes the notation u + z for u z,
u 2 z.)
b) We say that f is incrementally right continuous if, for all
2
z R we have (denoting z' = (s',t'))
lim |A z (f) = lim If(s',t')-f(s',t)-f(s,t')+f(s,t)
z'-z (s',t')*(s,t)
z'"z s'"s
t'St
= 0.
Remarks.
1) The limits are path-independent: in particular, in (a), this
limit includes the path where u z along a vertical or
horizontal path.
2) In (b) if s' = s or t' = t, A zz,(f) = 0, so we can take the
inequalities s' s, t' Z t to be strict. The chosen
definition is simply to preserve symmetry in the limits in (a)
and (b).
3) When we say simply, "f is right continuous," without further
specification, it will always mean in the sense of (a).
4) If f is right continuous, then f is incrementally right
continuous. To see this, note that IA zzfj = f(s',t')-
f(s',t)-f(s,t')+f(s,t)| = If(s',t')-f(s,t)+f(s,t)-f(s',t)-
f(s,t')+f(s,t)I (adding and subtracting f(s,t))
S lf(s',t')-f(s,t)| + If(s',t)-f(s,t)j + Af(s,t')-f(s,t) .
Then f right continuous implies each of the three terms on the