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How would I find a convex function $f: \mathbb{R} \to \mathbb{R}$ such that $\partial f(0) = [0,1]$

A subdifferential is just the collection of vectors $w \in \mathbb{R}^n$ such that

$f(y) \geq f(x_0) + \langle w,y-x_0\rangle$, $\forall y \in \mathbb{R}^n$ and $\partial f(x_0) \neq \emptyset$.

I am unsure how to extend this definition to my question, and would appreciate any help.

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    $\begingroup$ Hint: What is the subdifferential of $|x|$ at $0$? $\endgroup$ – user99914 Feb 7 '15 at 5:29
  • $\begingroup$ This would be [-1,1] ? $\endgroup$ – yaweh Feb 7 '15 at 6:49
  • $\begingroup$ Yes, try to use $|x|$ to construct the function you want. $\endgroup$ – user99914 Feb 7 '15 at 6:59
  • $\begingroup$ Would $$ f(x)= \begin{cases} x^2,& \text{if } x\leq 0\\ x, & x > 0 \end{cases}$$ work? $\endgroup$ – yaweh Feb 7 '15 at 7:31
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    $\begingroup$ haha, definitely. I've familiarised myself with this now, not quite as confusing anymore, thanks =) $\endgroup$ – yaweh Feb 7 '15 at 8:15
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I think that

$$ f(x) = \begin{cases} 0,&\text{ if } x \leq 0 \\ x,&\text{ if } x > 0 \end{cases} $$

is a good example.

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