(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