Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionx→‖∇f(x)‖−1∇f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)⩽f(x)} can be expressed as the convex hull of the limits of type {N(xn)}, where {xn} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.