Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose

$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$

where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ is parallel to $\eta$ on $C$). Then one can easily show that $h(t)=u(f(t))$ is constant for every $C^1$ curve lying on the surface $C$, and therefore $u$ must be constant on $C$.

My questions is that how much the assumption $u \in C^1$ can be weakened. Can we reach a similar (and possibly weaker) conclusion if $u$ is only assumed to be in $W^{1,1}$ or $BV$? For instance if we only assume $u\in BV$ and $\frac{Du}{|Du|}\cdot \eta =1$, $|Du|$ a.e. on $C$, then can we conclude that $u^{-1}(C)$ is countable in the trace sense? (Assuming that $\frac{Du}{|Du|} \in (L_c^{\infty}(R^2))^n$ has a trace $T\in L^{\infty}(C)$. See http://www.math.northwestern.edu/~gqchen/10-Papers/ChenTorresZiemer.pdf)