$ 0 = \lim_{\varepsilon \rightarrow o^+} \int_{\Omega} \frac{ \chi( u + \varepsilon \xi > 0 ) - \chi(\{ u>0 \} ) }{\varepsilon}? $

Question

Let $\Omega $ be a domain, $u \in H^{1}(\Omega)$ and $\xi \in C^{\infty}_{0}(\{u > 0\})$ you can assume that $\{u > 0\}$ is an open set. I'd like to know if in this situation we can conclude that $$ 0 = \lim_{\varepsilon \rightarrow o^+} \int_{\Omega} \dfrac{ \chi( u + \varepsilon \xi >0 ) - \chi(\{ u>0 \} ) }{\varepsilon}? $$ Where $\chi(A) $ is the characteristc function of a set $A$. Or if can I assume another similar condition for that the limit above be true. This ask is motivated for instance by the question here. I've also read other things that motivate this suspicion.