Abstract
Let f 0 be a mapping of an open set R in n-space E n into m-space E m. Let us consider, along with f 0, all mappings f which are sufficiently good approximations to f 0. By the Weierstrass Approximation Theorem, there are such mappings f which are analytic; in fact, (see [5], Lemma 6) we may make f approximate to f 0 throughout R arbitrarily well, and if f 0 is r-smooth (i.e., has continuous partial derivatives of orders ≦r), r finite, we may make corresponding derivatives of fapproximate to those of f 0.
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Bibliography
E. E. Levi, I. problemi dei valori…, Memorie die Mat. e di Fis. d. Soc. Italiana d. Science, Serie Terza, 16 (1910), pp. 88–90.
M. Morse, Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc., 27 (1925), pp. 345–396.
M. Morse, The critical points of a function of n variables, Trans. Amer. Math. Soc., 33 (1931), pp. 72–91.
A. W. Tucker, Branched and folded coverings, Bull. Amer. Math. Soc., 42 (1936), pp. 859–862.
H. Whitney, Analytic extensions of differentiate functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), pp. 63–89.
H. Whitney, Differentiable manifolds, Ann. of Math., 37 (1936), pp. 645–680.
H. Whitney, Differentiability of the remainder term in Taylor’s formula, Duke Math. J., 10 (1943) pp. 153–158. (We note a correction. The lemma on p. 156 is not true. The following theorem should contain the hypothesis that fn is of class Cp).
H. Whitney, Differentiate even functions, ibid., pp. 159–160.
H. Whitney, The general type of singularity of a set of 2n − 1 smooth functions of n variables, ibid., pp. 161–172.
H. Whitney, The singularities of a smooth n-manifold in (2n−1)-space Ann. of Math., 45 (1944), pp. 247–293.
N. Z. Wolfsohn, On differentiate maps of Euclidean n-space into Euclidean m-space Harvard thesis, 1952. See the abstract in Bull. Amer. Math. Soc., 61 (1955), p. 171.
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© 1992 Birkhäuser Boston
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Whitney, H. (1992). On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane. In: Eells, J., Toledo, D. (eds) Hassler Whitney Collected Papers. Contemporary Mathematicians. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2972-8_27
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DOI: https://doi.org/10.1007/978-1-4612-2972-8_27
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