Suppose that $f = ι_{\mathbb{R}_+}$. Show that

Question

Suppose that $f = ι_{\mathbb{R}_+}$. Show that $0$ is in the boundary of dom f and that $∂f(0)$ is nonempty using the definition of the sub differential.

Any hints or suggestions is greatly appreciated.

Answer 1

By $\text{dom }f$, I suppose you mean the set on which $f$ is strictly less than $\infty$. Well, clearly in your case $\text{dom }f = \mathbb{R}_+$, and its boundary is $\{0\}$, which of course contains $0$. Finally, using the definition of subdifferentials, one computes \begin{eqnarray} \begin{split} \partial f(0) &:= \{v \in \mathbb{R} \text{ s.t }f(z) \ge f(x) + \langle v, z - 0\rangle\text{ }\forall z \in \mathbb{R}\} = \{v \in \mathbb{R} \text{ s.t }vz \le 0\text{ }\forall z \in \mathbb{R}_+\}\\ &= \{v \in \mathbb{R} : v \le 0\} \ne \emptyset. \end{split} \end{eqnarray}

Alternatively, you can use the theory of proximal operators to get \begin{equation} v = v - 0 \in \partial f(0) \iff 0 = \mathrm{prox}_f(v) = \mathrm{proj}_{\mathbb{R}_+}(v) = (v)_+ \iff v \le 0. \end{equation}