right tends to zero as (s',t') decreases to (s,t), so we have
lim If(s',t') f(s,t) = 0,
(s',t')+(s,t)
i.e., f is right continuous. I
Remark. We still have the same result if we extend f, Ifl by zero
2
outside R.
+
The next result concerns the existence of "one-sided" limits and
"limits at infinity."
Theorem 2.3.3. a) Let f: R2 E be a function with finite variation
Ifl. Then each of the following limits exists* at each z = (s,t) e R2:
1) f(s@,t+) lim f(s',t')
t'tt
) f(s ,t ) = lim f(s',t')
s'+ts
t'+t
Sf(s+,t_) = lim f(s',t').
t'+tt
Moreover, if f has bounded variation (i.e., if there exists M>0 such
2
that Ifl(s,t) < M for all (s,t) E R ), then each of the following
* Of course, on the axes, not all these limits make sense. It will
be understood that at each point we take limits from quadrants
where f is defined.