  We introduce real vector spaces composed of set-valued maps on an open set.
They are also complete metric spaces, lattices, commutative rings. The set of
differentiable functions is a dense subset of these spaces and the classical
gradient may be extended in these spaces as a closed operator. If a function f
belongs to the domain of such extension, then f is locally lipschitzian and the
values of extended gradient coincide with the values of Clarke's gradient.
However, unlike Clarke's gradient, our generalized gradient is a linear
operator.
