Confusion related to the gradient of the sum of a smooth and non-smooth function

Confusion related to the gradient of the sum of a smooth and non-smooth
function

I have this confusion related to gradient. Let my function
$f(x) = g(x) + h(x)$ where $g(x)$ is a differentiable function and $h(x)=
\lambda\|x\|_1$
$g(x)$ is differentiable but $\|x\|_1$ is not at $x=0$. I found out that
gradient of $f(x)$ for $x=0$ is
$$\operatorname{sign}(\operatorname{grad}(g(x)))\max(|\operatorname{grad}(g(x))|-\lambda,0)$$
I didn't get how this formula was derived. Any suggestions? I was actually
reading this paper where they have mentioned