Publication:
An elementary example of Sard's Theorem sharpness

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2022-02-17
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Abstract
In this note we define a C1 function F : [0, M] 2 → [0, 2] that satisfies that its set of critical values has positive measure. This function provides an example, easier than those that usually appear in the literature, of how the order of differentiability required in Sard’s Theorem cannot be improved.
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Matemáticas (Matemáticas) , Análisis matemático
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12 Matemáticas , 1202 Análisis y Análisis Funcional
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[1] J.S. Athreya, B. Reznick, J.T. Tyson. Cantor Set Arithmetic, The American Mathematical Monthly, 126 (1) (2019) 4–17. [2] S.M. Bates. Towards a precise smoothness hypothesis in Sard’s theorem Proc. Amer. Math. Soc. 117 (1) (1993) 279–283. [3] A.P. Morse. The behavior of a function on its critical set Ann. of Math. 40 (1939) 62–70. [4] A. Norton Functions not constant on fractal quasi-arcs of critical points Proc. Amer. Math. Soc. 106 (2) (1989) 397–405. [5] A. Sard. The measure of the critical values of differentiable maps Bull. Amer Math. Soc. 48 (1942) 883–890. [6] J. Shallit. Quickies Q785, Mag. Math. 64 (5) (1991) 351–357. [7] H. Whitney. A function not constant on a connected set of critical points Duke Math. J. 1 (1935) 514–517.
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https://hdl.handle.net/20.500.14352/71710
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