35.2. TEST FUNCTIONS AND WEAK DERIVATIVES 1233

for all φ ∈C∞c (Ω). Recall that f ∈ L1

loc (Ω) if f XK ∈ L1 (Ω) whenever K is compact.Example: δ x ∈D∗ (Ω) where δ x (φ)≡ φ (x).It will be observed from the above two examples and a little thought that D∗ (Ω) is

truly enormous. The derivative of a distribution will be defined in such a way that it agreeswith the usual notion of a derivative on those distributions which are also continuouslydifferentiable functions. With this in mind, let f be the restriction to Ω of a smooth functiondefined on Rn. Then Dxi f makes sense and for φ ∈C∞

c (Ω)

Dxi f (φ)≡∫

Ω

Dxi f (x)φ (x)dx =−∫

Ω

f Dxiφdx =− f (Dxiφ).

This motivates the following definition.

Definition 35.2.1 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). In this weak sense, the derivative of almost anything exists, even functions thatmay be discontinuous everywhere. However the notion of “derivative” is very weak, hencethe 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 35.2.2 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.

In proving Theorem 35.2.2 the following lemma is useful.

Lemma 35.2.3 Let T ∈D∗ (a,b) and suppose DT = 0. Then there exists a constant C suchthat

T (φ) =∫ b

aCφdx.