Suppose $v:[a,b]\to R$ is absolutely continuous ($W^{1,1}$) and its derivative $f:\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous.
Is the function v then bounded ?
My reasoning so far: f is continuous $\Rightarrow$ v'=f is bounded $\Rightarrow$ with x AC $\Rightarrow$ x is Lipschitz on a compact set
Is my reasoning correct. If not what property might I be missing ?
Even easier:
Note that every absolute continuous function is continuous. Since the domain is compact, then by Extreme value theorem it admits a minimum point and a maximum point. Hence it is bounded.