944 CHAPTER 26. INTEGRALS AND DERIVATIVES

Definition 26.3.2 For T ∈D∗ (Ω)

DxiT (φ)≡−T (Dxiφ).

Of course one can continue taking derivatives indefinitely. Thus,

Dxix j T ≡ Dxi

(Dx j T

)and it is clear that all mixed partial derivatives are equal because this holds for the functionsin C∞

c (Ω). Thus one can differentiate virtually anything, even functions that may be dis-continuous everywhere. However the notion of “derivative” is very weak, hence the name,“weak derivatives”.

Example: Let Ω = R and let

H (x)≡{

1 if x≥ 0,0 if x < 0.

ThenDH (φ) =−

∫H (x)φ

′ (x)dx = φ (0) = δ 0(φ).

Note that in this example, DH is not a function.What happens when D f is a function?

Theorem 26.3.3 Let Ω = (a,b) and suppose that f and D f are both in L1 (a,b). Then f isequal to a continuous function a.e., still denoted by f and

f (x) = f (a)+∫ x

aD f (t)dt.

The proof of Theorem 26.3.3 depends on the following lemma.

Lemma 26.3.4 Let T ∈D∗ (a,b) and suppose DT = 0. Then there exists a constant C suchthat

T (φ) =∫ b

aCφdx.

Proof: T (Dφ) = 0 for all φ ∈C∞c (a,b) from the definition of DT = 0. Let

φ 0 ∈C∞c (a,b) ,

∫ b

aφ 0 (x)dx = 1,

and let

ψφ (x) =∫ x

a[φ (t)−

(∫ b

aφ (y)dy

)φ 0 (t)]dt

for φ ∈C∞c (a,b). Thus ψφ ∈C∞

c (a,b) and

Dψφ = φ −(∫ b

aφ (y)dy

)φ 0.