  We construct a differentiable locally Lipschitz function $f$ in
$\mathbb{R}^{N}$ with the property that for every convex body $K\subset
\mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides
with the set $\partial_L f(\bar x)$ of limits of derivatives
$\{Df(x_n)\}_{n\geq 1}$ of sequences $\{x_n\}_{n\geq 1}$ converging to~$\bar
x$. The technique can be further refined to recover all compact connected
subsets with nonempty interior, disclosing an important difference between
differentiable and continuously differentiable functions. It stems out from our
approach that the class of these pathological functions contains an infinite
dimensional vector space and is dense in the space of all locally Lipschitz
functions for the uniform convergence.
