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It is well known that if $f : [0;1] \to {\mathbb R}$ is a nondecreasing function, then the set $E$ of points where $f$ is not differentiable has Lebesgue measure zero. Is there an example where $E$ is not countable ? .

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    Thanks for the references Dave.2012-03-29

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The Cantor function is a standard example.

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    Of course, where was my mind?2012-03-29
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You could find a whole chapter devoted to this topic, with lots of examples and counterexamples in the book http://books.google.fr/books/about/A_first_course_in_Sobolev_spaces.html?id=W3RLWwnY0RkC&redir_esc=y