I've been looking through the notes on Sobolev Spaces here: https://www.math.ucdavis.edu/~hunter/pdes/ch3.pdf
Proposition 3.22 states that if $u\in L^1_{loc}(\Omega)$ has weak derivative $\partial_i u \in L^1_{loc}(\Omega)$ then: $$\partial_i |u|=\begin{cases} \partial_i u & u> 0 \\ 0 & u=0 \\ -\partial_i u & u<0 \end{cases} $$
The notes then mention that an immediate corollary of this result is that: $$\partial_i u^+=\begin{cases} \partial_i u & u> 0 \\ 0 & u\leq0 \end{cases} $$
However, when I made the same computation, I arrived at: $$\partial_i u^+=\begin{cases} \partial_i u & u> 0 \\ \frac{1}{2}\partial_i u & u=0 \\ 0 & u<0 \end{cases} $$
Is my calculation correct? If so, are the two formulations equivalent?
It feels perhaps logical to me that they are, but $\{ x\in \Omega : u(x)=0 \}$ is not necessarily of zero Lebesgue measure. In that case, maybe it is then true that $\partial_i u=0$ almost everywhere, but I'm not sure how one would show this.
Any help would be greatly appreciated, thanks!