VarR(f) S Var (f) + VarR (f).
For the other inequality, if Q1 Q2 are any grids on R ,R2'
respectively, then Q U Q2 is a grid on R, and we have
VarR (f;Q) + VarR (f;Q) VarR(f;Q)
1 2
5 sup VarR(f;Q) = VarR (f).
Q=oXT
Since this inequality holds for any grid Q1 on R we have
sup VarR (f;Q1) + VarR2(f;Q2) S Var (f),
Q1 -Ox 1 2
I .e.,
Var (f) + VarR(f;Q ) < VarR(f).
Similarly, Q2 being arbitrary, we have on taking supremum for Q2.
VarR (f) + VarR (f) 5 VarR(f).
Putting the two together, we have the equality:
Var (f) + Var (f) VarR(f).
1 2
iv) Assume R1 = [(s,t),(s',t')] is contained in
R2 = [(q,r),(q',r')]. Then we have q S s < s' s q', and
r S t < t' S r' (see Figure 2-3). By the additivity property, we have
VarR (f) = VarR (f) + VarR (f) + Var (f) + VarR (f) + Var (f).
2 1 3 14 5 6
Since each term on the right is nonnegative, we have
VarR2 (f) VarR (f).
B? B