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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 - 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]