35.5. RADEMACHER’S THEOREM 1241

This proves the corollary.It may be interesting at this point to recall the definition of differentiability on Page

117. If you knew the above inequality held for ∇u having components in L1loc (Rn) , then at

Lebesgue points of ∇u, the above would imply Du(x) exists. This is exactly the approachtaken below.

35.5 Rademacher’s TheoremThe inequality of Corollary 35.4.2 can be extended to the case where u and u,i are inLp

loc (Rn) for p > n. This leads to an elegant proof of the differentiability a.e. of a Lip-

schitz continuous function as well as a more general theorem.

Theorem 35.5.1 Suppose u and all its weak partial derivatives, u,i are in Lploc (R

n). Thenthere exists a set of measure zero, E such that if x,y /∈ E then inequalities 35.4.2 and 35.4.1are both valid. Furthermore, u equals a continuous function a.e.

Proof: Let u ∈ Lploc (R

n) and ψk ∈C∞c (Rn) ,ψk ≥ 0, and ψk (z) = 1 for all z ∈ B(0,k).

Then it is routine to verify that

uψk, (uψk),i ∈ Lp(Rn).

Here is why:

(uψk),i (φ) ≡ −∫Rn

uψkφ ,idx

= −∫Rn

uψkφ ,idx−∫Rn

uψk,iφdx+∫Rn

uψk,iφdx

= −∫Rn

u(ψkφ),i dx+∫Rn

uψk,iφdx

=∫Rn

(u,iψk +uψk,i

)φdx

which shows(uψk),i = u,iψk +uψk,i

as expected.Let φ ε be a mollifier and consider

(uψk)ε ≡ uψk ∗φ ε .

By Lemma 35.3.5 on Page 1237,

(uψk)ε,i = (uψk),i ∗φ ε .

Therefore(uψk)ε,i→ (uψk),i in Lp (Rn) (35.5.3)

and(uψk)ε → uψk in Lp (Rn) (35.5.4)