# On the additive property of the subdifferential of lower semicontinuous functions

Let $$f:\mathbb R\to\mathbb R$$ be a lower semicontinuous function, we define the Fréchet subdifferential of $$f$$ at $$x\in\mathbb R$$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)-f(x)-Lv}{|v|}\geq0\right\}.$$ According to this paper, in the third page, authors claimed that this subdifferential satisfies a additive property, namely $$\partial^F(f+g)(x)\subset \partial^Ff(x)+\partial^Fg(x)\tag{*}$$ for any convex function $$g:\mathbb R\to\mathbb R$$ (here I choose the Banach space $$\mathbb R$$, then $$g$$ is automatically continuous and $$\partial^F g(x)\neq\emptyset$$ for all $$x\in\mathbb R$$.)

As usual, the summation in $$(*)$$ should be the Minkowski sum, hence in particular we have $$\emptyset+S=\emptyset$$ for all $$S\subset\mathbb R$$.

I'm trying to prove $$(*)$$, but here is a counterexample: Let $$f(x)=-|x|, g(x)=|x|$$ for all $$x\in\mathbb R$$, then $$g$$ is a convex function and $$\partial^Fg(0)=[-1,1]$$ and $$\partial^F(f+g)(0)=\{0\}$$; however, $$\partial^Ff(0)=\emptyset$$: if $$L\in\partial^Ff(0)$$, then $$\liminf_{v\to0}\frac{-|v|-Lv}{|v|}\geq0$$, i.e., $$-1-|L|\geq0$$, which is a contradiction. Now it seems that I've got a counterexample of $$(*)$$, but $$(*)$$ is used everywhere in literature, so what am I missing now?

In the same paper, at the end of the proof of Theorem 3.1, authors used an argument like this: if $$f+h$$ takes minimum at $$\bar x$$ with $$h$$ convex and $$\partial^F h(\bar x)\neq\emptyset$$, then $$0\in \partial^F(f+h)(\bar x)\subset \partial^Ff(\bar x)+\partial^Fh(\bar x);$$ combining this with $$\partial^Fh(\bar x)\subset (-\delta,\delta)$$ gives that $$\partial^Ff(\bar x)\cap(-\delta,\delta)\neq\emptyset$$. But my example above still indicates the failure of this argument.

Theorem 3.1 impiles the fact that $$\operatorname{dom}(\partial^Ff)$$ is dense in $$\mathbb R$$ if $$f:\mathbb R\to\mathbb R$$ is a lower semicontinuous function, which is a well-known result. So I must miss something here.

Thanks for any help!

In part (P3) of Definition 2.1 in the paper you linked, it is also required that $$g$$ be $$\partial$$-differentiable at $$x$$ (meaning that both $$\partial g(x)$$ and $$\partial(-g)(x)$$ are nonempty), which is not the case in your example (with $$\partial=\partial^F$$ and $$g(\cdot)=|\cdot|$$).
• Thanks for your answer! I understand now. I didn't realize the stronger condition of $\partial$-differentiable, which is also the reason why the authors require $p=2>1$ in the proof of Theorem 3.1, since $|x|^2$ is smoother than $|x|$. However, I've seen in another paper, in the proof of Theorem 7, Rockafellar invoked the ordinary Ekeland's variational principle, in which case $h(x)=A|x|(A>0)$ and he still used the property $(*)$. Is there any other particular reason in Rockafellar's proof? Nov 26, 2023 at 2:33