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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?

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    Nope. See http://www.math.tamu.edu/~tvogel/gallery/node17.html for instance.2011-03-11
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    You need all directional derivative to be continuous or similar conditions to prove that.2011-03-11
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    @Subramani: Please see Kumaresan's notes on A Conceptual Introduction to Multivariable Calculus.2011-03-12

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