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.