Var[(st t f) = Var[(s,t),(s.t.)]


 + Var[(st (s',t")]


(See Figure 2-2.)


1 S Si
 S s' st


 Figure 2-2 Additivity of the variation



 iv) If R1 C R2, Var (f) S VarR (f)
 1 2
 v) If f is right continuous (order sense), we can compute

 Var [zz'(f) using grids consisting of points with rational
 E zz 7
 coordinates (and hence we can take the supremum along a

 sequence of partitions).

Proof.

 i) Since Var [z,'(f;P) Z 0 for any partition P, we have


 Var [zz'](f) = sup Var z,z'](f;P) 0.
 [zz'] [zz']

 ii) If Var [zz](f) = + -, then for every N>0, there is a
 such that Varz,z By Lemma 2.2.2, there
partition PN such that Var .](f;P ) > N. By Lemma 2.2.2, there
 -N [z,z'-N