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I am neither aware fully nor have studied differential geometry, but i'd like to learn it if i get to know the answer for this question. I am asking this question based on the very superficial knowledge of differential geometry i got know reading wikipedia.

Does a manifold exist whose degree of differentiability is different at different points ?

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Yes, certainly. For example the solutions $C$ to the equation $y=|x|$ is a topological submanifold of $\mathbb R^2$. Away from the origin it's a $C^\infty$-manifold, meaning that for sufficiently small neighbourhoods $U$ of points, $U \cap C$ is $C^\infty$. But if you intersect $C$ with any neighbourhood of the origin, it's never even a $C^1$-manifold. So you can make sense of "degree of differentiability near a point". For abstract manifolds a sheafy language would be the most natural way to phrase your question.

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    than$k$s for your answer. Let's just say I hadn't had my morning coffee while writing that post!2011-04-08