We consider a Sobolev function $u\in W^{1,p}(\Omega)$ for some $p>1$ and a bounded domain $\Omega\in\mathbb{R}^n$. Assuming we have shown the inequality $$|\partial_s u(x)|=|\nabla u(x) \cdot s|\leq C$$ for a.e. $x\in B_R\Subset \Omega$ and any $s\in\mathbb{R}^n$ with $|s|=1$, where $C>0$ denotes some finite quantity (so we have a pointwise estimate for the directional derivative of $u$ in direction $s$). Does this in particular imply $$|\nabla u(x)|\leq C \quad (*)$$ for a.e. $x\in B_R\Subset \Omega$? I know that the choice $\hat{s}=\frac{\nabla u(x)}{|\nabla u(x)|}$ ($|\hat{s}|=1$) would yield that - however, with this choice $\hat{s}$ would depend on the spatial point $x$, and in the inequality above I think of the spatial direction $s$ as constant, i.e. s should be independent of $x$. So my question is: does this dependence on $x$ not pose an issue in the choice of $\hat{s}$ or is there another way to see the validity of $(*)$?
Thanks!
If $C$ is independent of $s$, then the claim follows: it is a consequence of the equivalence: $$ \|x\|_2 \le c \ \Leftrightarrow |x^Ts|\le c \quad\forall s: \|s\|_2\le 1. $$