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Let $U\subset \mathbb{R}^{n}$ be an open set and $f:U \to \mathbb{R}$ a continuous function which is piecewise $C^{1}$. This is: there is a partition of $U$ by (say, a finite number of) open sets $U_{\alpha}$, each with piecewise $C^{1}$ boundary, such that $f$ restricted to each of these is $C^{1}$.

My question is the following: (when) are the weak derivatives of $f$, $D_{i} f$, given by the piecewise defined functions $g_{i}|_{U_{\alpha}}=D_{x_{i}}(f|_{U_{\alpha}})$? Where can I find a proof if the case?

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    I am sorry that I couldn't understand your notation: What is $g_i$ here?2011-12-03
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    Are the Lakes of Wada a counterexample?2011-12-03
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    @Paul: maybe the notation is a bit dumb. I have add a parenthesis that hopefully makes it clear (remember that weak derivatives are well defined if specified up to a set of measure zero, so this assumption is implicit w.r.t the union of the $\partial U_{\alpha}$).2011-12-03
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    @ user7530: I don't have time to study that example right now (I have just look for "Lakes of Wada" at Wikipedia), but it probably goes beyond the assumptions of my problem. For example: is the boundary of these "lakes" piecewise $C^{1}$? Also: what function are you expecting to define over them?2011-12-03

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