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Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve

Published online by Cambridge University Press:  20 November 2018

Laurent Moonens
Laurent Moonens*
Affiliation:
Département de mathématique, Université catholique de Louvain, 1348 Louvain-la-neuve, Belgiume-mail: [email protected]
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Abstract

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We provide a simple example showing that the tangential derivative of a continuous function $\phi $ can vanish everywhere along a curve while the variation of $\phi $ along this curve is nonzero. We give additional regularity conditions on the curve and/or the function that prevent this from happening.

Keywords

26A2428A15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 706 - 715
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] De Pauw, T., Moonens, L., and Pfeffer, W. F., Charges in middle dimensions. J. Math. Pures Appl. (9) 92(2009), no. 1, 86–112.Google Scholar
[2] Federer, H., Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, 153, Springer-Verlag, New York, 1969.Google Scholar
[3] Pfeffer, W. F., The Riemann approach to integration. Local geometric theory. Cambridge Tracts in Mathematics, 109, Cambridge University Press, Cambridge, 1993.Google Scholar
[4] Saks, S., Theory of the integral. Stechert, New York, 1937.Google Scholar
[5] Whitney, H., A function not constant on a connected set of critical points. Duke Math. J. 1(1935), no. 4, 514–517. doi:10.1215/S0012-7094-35-00138-7Google Scholar