Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ?
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3What do you know about differentiability of convex functions? – Daniel Fischer May 19 '15 at 13:51
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Not much (or say nothing), any readings or references would be appreciated. – incas May 19 '15 at 14:14
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2One characterisation of convexity is that the difference quotients $$\frac{f(y) - f(x)}{y-x}$$ where $x\neq y$ are increasing in each variable when the other is fixed. Can you see how that relates to differentiability? – Daniel Fischer May 19 '15 at 14:29
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this would give me existence of right and left derivative everywhere ? moreover left derivative is less or equal then right derivative. – incas May 19 '15 at 14:35
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Ok maybe I have better, the function will be lipischitz in every compact set. So that it will be absolutely continuous. – incas May 19 '15 at 14:40
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Right, you can argue with that. That's probably the shortest way. – Daniel Fischer May 19 '15 at 14:42
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Thx very much, very helpful. – incas May 19 '15 at 14:43
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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.
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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.
