Let $u, v \in L^p(\Omega)$ where $\Omega\subset \mathbb{R}^d$, and $S:\mathbb{R}\rightarrow \mathbb{R}$ is the sigmoid function, $$S(x) = \frac{1}{1+e^{-x}}.$$
I want to know whether I can apply mean value theorem inside of the $L^p$-norm in point-wise manner, i.e., there exists $w\in L^p(\Omega)$ such that $$||S(u) - S(v)|| = ||S'(w)(u-v)||$$ and $w$ lies between $u$ and $v$.
I'm not sure whether I can apply mean-value theorem point-wisely when the point-wise value of $u$ and $v$ are not well-defined.