  Let $\Omega$ be a bounded domain in $\mathbb{C}$ such that $\partial \Omega$
does not contain isolated points. Let $R(\Omega)$ be the space of uniform
limits on $\overline{\Omega}$ of rational functions with poles off
$\overline{\Omega}$, endowed with the supremum norm. We prove that either
generically all functions $f$ in $R(\Omega)$ satisfy % $$ \limsup_{\substack{z
\to z_0 z \in \partial \Omega}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = +
\infty $$ for every $z_0 \in \partial \Omega$ or no such function in
$R(\Omega)$ meets this requirement. In the first case, the generic function $f
\in R(\Omega)$ is nowhere differentiable on $\partial \Omega$ with respect to
the position. We give specific examples where each case of the previous
dichotomy holds. We also extend the previous result to unbounded domains.
