I would like to know if this two definitions for Sobolev Spaces are equivalent: $H_2^m([a,b]):=\{f: f',...,f^{(m-1)} \text{absolutely continuous}, \int _a^b (f^{(m)})^2dx<\infty \}$
and $H_2^m([a,b])$ contains all functions $f$ such that $D^\mu f \in L_2([a,b])$ for $\mu=0,1,...,m$
The first definition requires the $m$th derivative to be square integrable, while in the second one all derivatives (up to order m) must belong to $L_2([a,b])$ and no continuity is required. I guess it has to do with the fact that absolute continuity is required, but I want to be sure that both are valid. Any hint will be greatly appreciated.