coordinates such that
IVar, '(fQ ) VarZZ](f;Q ) <
Lzz'] ar[z,z'] C t
hence
Var ,z (f;Q) > Var[zz(f;Q ) (a a .
[z,z] [zz E 2 2 2
c arbitrary => sup Var[ (f,Q) = Var ,](f). I
Q=z,z'] [z,z']
Q=oxT
Qrational
An important property of the variation in one dimension is that a
function of finite variation f is right continuous if and only if
lim Var (f) = 0 for all s in the domain of f. Unfortunately,
s 4s [s,s']
we do not have this equivalence in two dimensions without additional
assumptions about f. We do have one implication, however.
Theorem 2.2.6. Let f: R + E be a function with Var (f) < for
+ R
every bounded rectangle R. If f is right continuous, then for every
z, z', u in R2 with z < z' < u, we have
Var (f) = lim Var (f).
(Note: The notation u+,z' means u z', u > z'.)
Proof. We divide the region outside [z,z'3 and inside [z,u] into
three parts, labeled R1, R2, R (see Figure 2-4). The proof consists
r u
R3 R1
t
z
R2
t
zs s' p
Figure 2-4 Decomposition of [z,u]