Bounded derivative on interval implies almost everywhere boundedness?

Question

Consider a sequence $(f_n)$ with $f_n$ in $L^p(I)$ where $I=(a,b)$ and let $h$ be a constant. Does $\| f'_n \|_p < h$ imply that $|f_n| < k$ almost everywhere for a suitable $k$?

Answer 1

The $p$-norm allows $f'$ to be undefined on a set of measure zero, so $f$ can have jumps as big as you like on a set of measure zero since $L^p$ permits such things. We can thus arrange a jump so large that $|f_n| > k$ for any $k$ you specify.