This is related to This Question when I want to show that if a function $u$ has a strong derivative then the function $u^-:=\min \{0,u\}$ has a weak one. This is my question:
Let $I=(a,b)\subset \mathbb{R}$ and $u:I\to \mathbb{R}$ be a continuously differentiable function and $\phi\in C^\infty_c(I)$ (smooth compactly supported function), then by integration by parts $$\int_a^bu(t)\phi'(t)\,dt=-\int_a^bu'(t)\phi(t)\,dt.$$ Let $I_1=\{t\in I: u(t)\leq 0 \}$. Do we have $$\int_{I_1}u(t)\phi'(t)\,dt=-\int_{I_1}u'(t)\phi(t)\,dt$$ and why ?