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Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve

  Published:2011-03-05
 Printed: Dec 2011
  • Laurent Moonens,
    Département de mathématique, Université catholique de Louvain, 1348 Louvain-la-neuve, Belgium
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Abstract

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.
MSC Classifications: 26A24, 28A15 show english descriptions Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Abstract differentiation theory, differentiation of set functions [See also 26A24]
26A24 - Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
28A15 - Abstract differentiation theory, differentiation of set functions [See also 26A24]
 

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