Implications of the constant rank constraint qualification

An example in which CRCQ holds. The left figure shows a set S(u) defined by 3 constraints. The normal cone NS(u)(x) is the cone generated by the two dashed halflines originated from the point x. The red curve indicates the neighborhood X. The normal cone NS(u)(x) has four nonempty faces, and the family {I(x′, u), x′ ∈ S(u)∩X} contains four elements {1, 3},{1},{3} and ∅, each of which corresponds to a face of NS(u)(x). E.g., {1, 3} corresponds to the cone NS(u)(x) itself, and ∅ corresponds to the vertex of NS(u)(x).