35.3. WEAK DERIVATIVES IN Lploc 1235

for all φ ∈C∞c (a,b). Thus∫ b

a{(

f (x)−∫ x

aD f (t)dt

)−C}φ (x)dx = 0

for all φ ∈C∞c (a,b). It follows from Lemma 35.3.3 in the next section that

f (x)−∫ x

aD f (t)dt−C = 0 a.e. x.

Thus let f (a) =C and write

f (x) = f (a)+∫ x

aD f (t)dt.

This proves Theorem 35.2.2.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 where the derivative is taken in the sense of a limit ofdifference quotients and f ′ (x) = D f (x). This raises an interesting question. Suppose f iscontinuous 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. To see an example, consider Problem 4 on Page 970 which gives anexample of a function which is continuous on [0,1], has a zero derivative for a.e. x butclimbs from 0 to 1 on [0,1]. Thus this function is not recovered from integrating its classicalderivative.

In summary, if the notion of weak derivative is used, one can at least give meaning tothe derivative of almost anything, the mixed partial derivatives are always equal, and, inone dimension, one can recover the function from integrating its derivative. None of theseclaims are true for the classical derivative. Thus weak derivatives are convenient and ruleout pathologies.

35.3 Weak Derivatives In Lploc

Definition 35.3.1 Let U be an open set in Rn. f ∈ Lploc (U) if f XK ∈ Lp whenever K is a

compact subset of U.

Definition 35.3.2 For α = (k1, · · · ,kn) where the ki are nonnegative integers, define

|α| ≡n

∑i=1|kxi |, Dα f (x)≡ ∂ |α| f (x)

∂xk11 ∂xk2

2 · · ·∂xknn.