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dif9-page4 - 44444 Remarks i The sublemma shows that the...

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Unformatted text preview: 44444 Remarks.-----------------------------------* i) The sublemma shows that the mapping i :M----------L F (M) above is indeed an inclusion. If p $ q are distinct points of M, then there is a smooth function h on M such that [ i (p)](h) = h(p) = 1 $ [ i (q)](h) = h(q) = 0. ii) We can extend a derivation D at a point p on functions f defined only in a neighborhood U of p by taking a smooth function h on M such that h is zero outside U and constant 1 in a neighborhood of p and putting D(f) := ~ D(f), where ( ~ f(x)h(x) for x e U f(x) = { for x m U. 9 By lemma 2 this extension of D is correctly defined. n Lemma 3. Let f:B----------L R be a smooth function defined on an open ball B C R-------------------------around the origin. Then there exist smooth functions g 1 < i < n on B such that i n f( x ) = f( ) + S x g ( x ) for x = (x ,. ..,x ) e B i i 1 n i=1 and d f g ( ) =--------------------( ). i d x i 4...
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