938 CHAPTER 26. INTEGRALS AND DERIVATIVES

26.2 Absolutely Continuous FunctionsDefinition 26.2.1 Let [a,b] be a closed and bounded interval and let F : [a,b]→ R. ThenF is said to be absolutely continuous if for every ε > 0 there exists δ > 0 such that if∑

mi=1 |yi− xi|< δ where the intervals (xi,yi) are non-overlapping, then

m

∑i=1|F (yi)−F (xi)|< ε.

Definition 26.2.2 A finite subset, P of [a,b] is called a partition of [x,y] ⊆ [a,b] if P ={x0,x1, · · · ,xn} where

x = x0 < x1 < · · · ,< xn = y.

For f : [a,b]→ R and P = {x0,x1, · · · ,xn} define

VP [x,y]≡n

∑i=1| f (xi)− f (xi−1)| .

Denoting by P [x,y] the set of all partitions of [x,y] define

V [x,y]≡ supP∈P[x,y]

VP [x,y] .

For simplicity, V [a,x] will be denoted by V (x) . It is called the total variation of the func-tion, f .

There are some simple facts about the total variation of an absolutely continuous func-tion, f which are contained in the next lemma.

Lemma 26.2.3 Let f be an absolutely continuous function defined on [a,b] and let V beits total variation function as described above. Then V is an increasing bounded function.Also if P and Q are two partitions of [x,y] with P ⊆ Q, then VP [x,y] ≤ VQ [x,y] and if[x,y]⊆ [z,w] ,

V [x,y]≤V [z,w] (26.2.6)

If P = {x0,x1, · · · ,xn} is a partition of [x,y] , then

V [x,y] =n

∑i=1

V [xi,xi−1] . (26.2.7)

Also if y > x,V (y)−V (x)≥ | f (y)− f (x)| (26.2.8)

and the function, x→ V (x)− f (x) is increasing. The total variation function, V is abso-lutely continuous.

Proof: The claim that V is increasing is obvious as is the next claim about P ⊆ Qleading to VP [x,y] ≤ VQ [x,y] . To verify this, simply add in one point at a time and verifythat from the triangle inequality, the sum involved gets no smaller. The claim that V is