On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane
On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane
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DOI：
10.1007/978-1-4612-2972-8_27
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发表时间：
1955-11
影响因子：
4.9
通讯作者：
H. Whitney
中科院分区：
文献类型：
--
作者：
H. Whitney
设f 0是一个开集Rinn-spaceEnintom-spaceEm的映射。让我们沿着f 0考虑所有映射f，它们是f 0的充分好的近似。根据Weierstrass逼近定理，有这样的映射f是解析的;事实上，（见[5]，引理6）我们可以使f通过R任意好地逼近f 0，并且f 0是r-光滑的（即，有连续的阶为f（r），r有限的偏导数，我们可以使相应的导数f逼近于f（0）的导数。
Letf0be a mapping of an open setRinn-spaceEnintom-spaceEm. Let us consider, along withf0, all mappingsfwhich are sufficiently good approximations tof0. By the Weierstrass Approximation Theorem, there are such mappingsfwhich are analytic; in fact, (see [5], Lemma 6) we may makefapproximate tof0throughoutRarbitrarily well, and iff0isr-smooth (i.e., has continuous partial derivatives of orders ≦r),rfinite, we may make corresponding derivatives offapproximate to those off0.