Is convex subdifferential always non-empty?

Question

Let $f:X\to \mathbb{R}\cup\{+\infty\}$ be a proper, convex and l.s.c. function on a Banach space $X$. The mapping $\partial f:X\to 2^{X^*}$ defined by $$\partial f(x)=\{x^*\in X^*: (x^*, v-x)\le f(v)-f(x)\,\,\mbox{for all}\, v\in X\}$$ is called the convex subdifferential of $f$. I was thinking: what if $f(x_0)=\infty$? Since $f$ is proper, then there exists at least one element $v\in X$ such that $f(v)\neq\infty$. It follows that $x^*=-\infty$ for all $x\in X$ if I consider $\partial f(x_{0})$. Does it have any sense? Natural question emerges: is always $\partial f\neq\emptyset$?

Answer 1

It might be easier to consider the real case $X = \mathbb R$ first. Then, if $f(x_0) = \infty$ and $f(v) < \infty$, you never have $$\mathbb R \ni (x^*, v - x_0) \le f(v) - f(x_0) = -\infty,$$ hence $\partial f(x_0) = \emptyset$.

Even if $f(x_0) \in \mathbb R$, the subdifferential might be empty, consider $f(x) = -\sqrt{1-x^2}$ for $x \in [-1,1]$ and $f(x) = +\infty$ otherwise, at the points $x = \pm 1$.