942 CHAPTER 26. INTEGRALS AND DERIVATIVES

Now f ′ exists and is in L1 becasue f = V − (V − f ) and V and V − f have derivatives inL1. Therefore, (V − f )′ =V ′− f ′ and so the above reduces to

f (x)− f (a) =∫ x

af ′ (t)dt.

This proves one half of the theorem.Now suppose f ′ ∈ L1 and f (x) = f (a)+

∫ xa f ′ (t)dt. It is necessary to verify that f is

absolutely continuous. But this follows easily from Lemma 11.5.2 on Page 256 which im-plies that a single function, f ′ is uniformly integrable. This lemma implies that if ∑i |yi− xi|is sufficiently small then

∑i

∣∣∣∣∫ yi

xi

f ′ (t)dt∣∣∣∣= ∑

i| f (yi)− f (xi)|< ε.

The following simple corollary is a case of Rademacher’s theorem.

Corollary 26.2.6 Suppose f : [a,b]→ R is Lipschitz continuous,

| f (x)− f (y)| ≤ K |x− y| .

Then f ′ (x) exists a.e. and

f (x) = f (a)+∫ x

af ′ (t)dt.

Proof: It is easy to see that f is absolutely continuous. Therefore, Theorem 26.2.5applies.

26.3 Weak DerivativesA related concept is that of weak derivatives. Let Ω ⊆ Rn. A distribution on Ω is definedto be a linear functional on C∞

c (Ω), called the space of test functions. The space of all suchlinear functionals will be denoted by D∗ (Ω) . Actually, more is sometimes done here. Oneimposes a topology on C∞

c (Ω) making it into a topological vector space, and when this hasbeen done, D ′ (Ω) is defined as the dual space of this topological vector space. To see this,consult the book by Yosida [127] or the book by Rudin [114].

Example: The space L1loc (Ω) may be considered as a subset of D∗ (Ω) as follows.

f (φ)≡∫

Ω

f (x)φ (x)dx

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

loc (Ω) if f XK ∈ L1 (Ω) whenever K is compact.The following lemma is the main result which makes this identification possible.

Lemma 26.3.1 Suppose f ∈ L1loc (Rn) and suppose∫

f φdx = 0

for all φ ∈C∞c (Rn). Then f (x) = 0 a.e. x.