subdifferential of a linear function which has a non-negative domain

Question

I am trying to find the sub-derivative of the function $f(x) = \lambda x, x\ge0$. I am familiar of the sub-derivative of the absolute value but not sure how to find the sub-derivative of this function at $x=0$. Applying the same logic of the absolute value sub-derivative, I got the following expression:

$$\partial f(x)|_{x=0} = k, k\in(-\infty, \infty) - \{\lambda\}$$

My reasoning is that you can draw a line that touches $f(0)$ with 360 degrees rotations (except when the slopes equals to $\lambda$). Am I right? If yes, how to generalize it to a function with a domain $\in \mathbb{R}^n$?

Answer 1

The derivative at $0$ is not defined since right and left limits are not equal. Left limit does not exist, since the function is not defined for values less than zero. Right limit equals to $\lambda$.