exists a grid Q refining P ; by the remark following Definition 2.2.4,

we have Var ,(f;Q) a Var ,(f;P ) > N. Thus, if
 [z,z ] [z,z ] -N
Var[z,z ](f) = + -, for any N>O there exists a grid QN such that

Var z, (f;Q N) > N, i.e., Var [zz] (f) = sup Var zz](f;Q).
 [zz N [z.z ] Ez P .Z
 'IR
 Q=aox-
Similarly, if Varz,z ](f) = a<-, then for every E>O, there is a

partition P such that Var z (f,P ) > a-E. Again, taking a grid
 -E [z,z'] -E
Q refining P we have Var[ ,z(f;Q ) > Var z,'(f;P ) > a-c.
C -C [zz ] E [z,z'] -C
e arbitrary => sup Var [ ,(f;Q) 2 a = Varz ,z(f). The other
 QcP [z,z ] [z,z ']
 QEPR
 Q-oXT

inequality is evident, so we have Var[z,z'(f) = sup Varz,z(f;Q).

 Q=oxt
(Note: From now on, we shall compute variations using grids.)

 iii) We shall prove the first equality; the proof of the second is

completely analogous.

 Denote R = [(s,t),(s',t')], R2 = [(s',t),(s",t')], R = R 1 R2.

For any grid Q = ox- on R, we can add the point s' to o to get a

refinement Q' = Q U Q2, where Q is a grid on R Q2 is a grid

on R2. Then


 VarR(f;Q) S VarR(f;Q') = VarR (f;Q1) + VarR (f;Q)
 R R2

 5 sup VarR (f;Q1) + sup VarR (f;Q2)
 Q 1V Q( 2 2

 = Var (f) + Var (f).
 1 P2


Taking supremum on the left, we get