I want to find a subdifferential of function $f(x) = |c^Tx|$, where $x \in \mathbb{R}^n$.
I know that if $h(x) = f(Ax + b)$ then $\partial{h(x)} = A^T\partial{f(Ax+b)}$, which is exactly my case. Assume $g(x) = c^Tx$, hence, $f(x) = |g(x)|$. Which means that
$$\partial{f(x)} = c \ \partial{g(x)} = c \cdot c$$
Since $c$ is a column-vector $c \cdot c$ doesn't make sense for me. I may have missed something. Could somebody give me a hint?
We know that for $x\neq 0$, $f$ is differentiable. This implies that, for such $x$, the subdifferential is just $\{\nabla f(x)\}$. To find the subdifferential at $x=0$, you can use the result you mention. To do this, observe that, at $x=0$, $$\partial|x| = \{v: |x| \geq v \cdot x\} = \{v:|v|\leq 1\} = \overline{B}_1(0).$$