"limits of infinity" exist:
1') f(s ,) = llm f(s',t')
s'+ns
2') f(s ,-) = llm f(s',t')
t'+ts
3') f(=,t ) = lim f(s',t')
t'44t
4') f(m,t ) = lim f(s',t'), and especially
t '++t
5') f(m) = lim f(s',t') exists.
s',t't~
b) If, moreover, f is right continuous, then the one-sided limits
along the vertical and horizontal paths f(s,-), f(-t) are equal to the
following.
i) lim f(,t) = lim f(s,') = f(s+,t+) (right limits)
s'4~s t'++t
ii) lim f(-,t) = f(s-,t+), and
s'tts
lim f(-,t) = f(W,t+) if Ifl is bounded.
s 't
iii) lim f(s,-) = f(s+,t-), and
t'tt
lim f(s,-) = f(s+,-) if Ifl is bounded.
t'+
Remarks. 1) Here f is defined on R2: if we wish to use an f defined
also on the "boundary at infinity," the above limits will be denoted
with the symbol -- in place of ".