Let us assume that directional derivative of a function $f$ exists at a point $p$ (i.e.,$ D_v(f)$) for all vectors $v \in \mathbb{R}^{n}$. Does it imply that the function is differentiable?
directional derivative in a manifold
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calculus
analysis
differential-geometry
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4Nope. See http://www.math.tamu.edu/~tvogel/gallery/node17.html for instance. – 2011-03-11
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2You need all directional derivative to be continuous or similar conditions to prove that. – 2011-03-11
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0@Subramani: Please see Kumaresan's notes on A Conceptual Introduction to Multivariable Calculus. – 2011-03-12