is a convex continuous function absolutely continuous

Question

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ?

thank you.

Answer 1

This is true. In fact, it is locally in $W^{1,p}$ for any $p\in [1,\infty]$ because a convex function is locally Lipschitz.

Answer 2

Yes, this is true. A convex function is always continuous and actually locally Lipschitz. @Yes already provided you a nice proof in 1-D but let me point out that this result also holds in high-dimensions.

That is, given $f: \mathbb R^N\to \mathbb R$ is convex, we have $f$ is locally Lipschitz on $\mathbb R^N$. You can find proof at page 236 in this book, and also some more results of Convex function regarding to Sobolev spaces.