If a real function $f\colon[a,b]\to\mathbb{R}$ is differentiable and its derivative $f'$ is zero, then $f$ is constant. Does this result still hold when $f$ has a weak derivative?
Explicitly, suppose $f\colon[a,b]\to\mathbb{R}$ is an integrable function such that its distributional derivative $Df$ is zero. Does this mean that $f$ is constant?
The following corollary of the celebrated Du Boys-Reymond Lemma holds true.
Corollary. If $u \in L^1_{\mathrm{loc}}(a,b)$ is such that $$\int_a^b u(x) \varphi'(x)\, dx=0$$ for every $\varphi \in C_0^\infty(a,b)$, then $u$ is almost everywhere constant.
I wrote the statement of B. Dacorogna, Introduction au calcul des variations.