1254 CHAPTER 36. INTEGRATION ON MANIFOLDS

Furthermore, f is also a differentiable function2 and

D f (x) = ∑{DψU (x) : x ∈U ∈ C} .

Proof: Let p be a limit point of ∪C and let W be an open set which intersects onlyfinitely many sets of C. Then p must be a limit point of one of these sets. It followsp ∈ ∪

{H : H ∈ C

}and so ∪C ⊆ ∪

{H : H ∈ C

}. The inclusion in the other direction is

obvious.Now consider the second assertion. Letting x ∈ Rn, there exists an open set, W inter-

secting only finitely many open sets of C, U1,U2, · · · ,Um. Then for all y ∈W,

f (y) =m

∑i=1

ψUi(y)

and so the desired result is obvious. It merely says that a finite sum of differentiable func-tions is differentiable. Recall the following definition.

Definition 36.1.4 Let K be a closed subset of an open set, U. K ≺ f ≺U if f is continuous,has values in [0,1] , equals 1 on K, and has compact support contained in U.

Lemma 36.1.5 Let U be a bounded open set and let K be a closed subset of U. Then thereexist an open set, W, such that W ⊆W ⊆U and a function, f ∈C∞

c (U) such that K≺ f ≺U.

Proof: The set, K is compact so is at a positive distance from UC. Let

W ≡{

x : dist(x,K)< 3−1 dist(K,UC)} .

Also letW1 ≡

{x : dist(x,K)< 2−1 dist

(K,UC)}

Then it is clearK ⊆W ⊆W ⊆W1 ⊆W1 ⊆U

Now consider the function,

h(x)≡dist(x,WC

1

)dist(x,WC

1

)+dist

(x,W

)Since W is compact it is at a positive distance from WC

1 and so h is a well defined continuousfunction which has compact support contained in W 1, equals 1 on W, and has values in[0,1] . Now let φ k be a mollifier. Letting

k−1 < min(dist(K,WC) ,2−1 dist

(W 1,UC)) ,

it follows that for such k,the function, h∗φ k ∈C∞c (U) , has values in [0,1] , and equals 1 on

K. Let f = h∗φ k.The above lemma is used repeatedly in the following.

2If each ψU were only continuous, one could conclude f is continuous. Here the main interest is differentiable.