946 CHAPTER 26. INTEGRALS AND DERIVATIVES

Theorem 35.2.2 says that

f (x) = f (a)+∫ x

aD f (t)dt

whenever it makes sense to write∫ x

a D f (t)dt, if D f is interpreted as a weak derivative.Somehow, this is the way it ought to be. It follows from the fundamental theorem ofcalculus that f ′ (x) exists for a.e. x in the classical sense where the derivative is taken inthe sense of a limit of difference quotients and f ′ (x) = D f (x). This raises an interestingquestion. Suppose f is continuous on [a,b] and f ′ (x) exists in the classical sense for a.e.x. Does it follow that

f (x) = f (a)+∫ x

af ′ (t)dt?

The answer is no. You can build such an example from the Cantor function which isincreasing and has a derivative a.e. which equals 0 a.e. and yet climbs from 0 to 1. Thusthis function is not recovered from integrating its classical derivative. Thus, in a sense weakderivatives are more agreeable than the classical ones.

26.4 Lipschitz FunctionsDefinition 26.4.1 A function f : [a,b]→ R is Lipschitz if there is a constant K such thatfor all x,y,

| f (x)− f (y)| ≤ K |x− y| .

More generally, f is Lipschitz on a subset of Rn if for all x,y in this set,

|f(x)− f(y)| ≤ K |x−y| .

Lemma 26.4.2 Suppose f : [a,b]→ R is Lipschitz continuous and increasing. Then f ′

exists a.e., is in L1 ([a,b]) , and

f (x) = f (a)+∫ x

af ′ (t)dt.

If f : R→ R is Lipschitz, then it is in L1loc (R).

Proof: The Dini derivates are defined as follows.

D+ f (x) ≡ lim suph→0+

f (x+h)− f (x)h

, D+ f (x)≡ lim infh→0+

f (x+h)− f (x)h

D− f (x) ≡ lim suph→0+

f (x)− f (x−h)h

,D− f (x)≡ lim infh→0+

f (x)− f (x−h)h

For convenience, just let f equal f (a) for x < a and equal f (b) for x > b. Let (a,b) be anopen interval and let

Nab ≡{

x ∈ (a,b) : D+ f (x)> q > p > D+ f (x)}