  We introduce a real vector space composed of set-valued maps on an open set X
and note it by S. It is a complete metric space and a complete lattice. The set
of continuous functions on X is dense in S as in a metric space and as in a
lattice. Thus the constructed space plays the same role for the space of
continuous functions with uniform convergence as the field of reals plays for
the field of rationals. The classical gradient may be extended in the space S
as a close operator. If a function f belongs to its domain then f is locally
lipschitzian and the values of our gradient coincide with the values of
Clarke's gradient. However, unlike Clarke's gradient, our gradient is a linear
operator.
