TITLE: Does it exist a function for which the derivative changes sign more than countably many times?
QUESTION [10 upvotes]: Does there exist any function $f \in C^2[0,1]$; $f: [0,1] \mapsto [0,1]$, for which the derivative changes sign more than countably many times?

REPLY [2 votes]: How about a function $f'$ whose zero set is the Cantor set.  And on complementary intervals with length $3^{-k}$, $k$ odd, the function is positive, while on complementary intervals with $3^{-k}$, $k$ even, the function is negative.  Then could we say $f'$ "changes sign" at each point of the Cantor set (since there are positive values and negative values clustering at that point)?