Abstract
In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set
G\subset \mathbb{R}^{2}
we construct a differentiable function
f:G\to\mathbb{R}
for which there exists an open set
\Omega_{1}\subset\mathbb{R}^{2}
such that
\nabla f(\mathbf{p})\in \Omega_{1}
for a
\mathbf{p}\in G
but
\nabla f(\mathbf{q})\not\in\Omega_{1}
for almost every
\mathbf{q}\in G
. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.