Properties of the Subdifferential on $\mathbb{R}^n$: Minimisers, Convexity and Closedness, Sums and Positive Multiples

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$, let $f,g:C\to\mathbb{R}$ be functions, and let $x\in C$, with subdifferentials $\partial f(x)$, $\partial g(x)$ and $\partial(f+g)(x)$ of $f$, $g$ and $f+g:C\to\mathbb{R}$, $y\mapsto f(y)+g(y)$.

(i) Minimisers. $f(x)\le f(y)$ for every $y\in C$ if and only if $0\in\partial f(x)$.

(ii) Convexity. $\partial f(x)$ is a convex subset of $\mathbb{R}^{n}$.

(iii) Closedness. $\partial f(x)$ is closed.

(iv) Sums.

$$\{a+a':\ a\in\partial f(x),\ a'\in\partial g(x)\}\subseteq\partial(f+g)(x).$$

(v) Positive multiples. For every real $\lambda>0$, the subdifferential of $\lambda f:C\to\mathbb{R}$, $y\mapsto\lambda f(y)$, at $x$ is

$$\partial(\lambda f)(x)=\{\lambda a:\ a\in\partial f(x)\}.$$