2) In general, a function of finite variation can have eight
different limits at a point: the four "quadrantal" limits from part
(a) of the statement, plus the four one-sided limits along the
vertical and horizontal paths. Part (b) of the statements says that
if f is right continuous, the one-sided limits can be incorporated
into the quadrantal limits, so there are only four distinct limits at
a point (s,t) (at most). The first part of (b) says that both right
limits along the vertical and horizontal paths are equal to limit (1),
the second part says the left limit along the horizontal path is equal
to limit (2), and the third part says the remaining left limit is
equal to limit (4), giving the division of the plane shown in Figure
2-10. There are analogous considerations for the "limits at
infinity."
I
r
I
(s,t) (1) (2) (s,t)
(s t) (s t)
I
(3) (4)
Figure 2-10 The four "quadrants"