Geodesic
On a~property of the subdifferential
Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 63-78

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Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton–Jacobi equations to prove the equivalence of various types of definitions of generalized solutions. 
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     title = {On a~property of the subdifferential},
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A. I. Subbotin. On a~property of the subdifferential. Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 63-78. http://geodesic.mathdoc.fr/item/SM_1993_74_1_a5/