Corrections to second edition

But, arguing as in the previous paragraph, we can show that ∫ |x−y|<2 ∂k [( 1− η ( |x− y| )) ∂jK(x− y) ] |x− y| dy ≤ C( ), where C( ) → 0 as → 0, so |v(k)(x) − ∂kv (x)| ≤ C( ) sup y∈Ω |f(y)− f(x)| |y − x| . But f ∈ C(Ω) ensures that the supremum is finite and continuous in x ∈ Ω, so we conclude that ∂kv → v(k) uniformly on compact neighborhoods of x as → 0. In particular, v ∈ C(Ω), and hence u ∈ C(Ω). ♠ p. 118. “In the next section” should be “In the next two subsections,” and C(Ω) should be replaced by C(Ω) ∩C1(Ω) right after (33) and in the last paragraph of this subsection.