In mathematics, particularly in the study of manifolds and differential topology, a "cusp" generally refers to a type of singular point or feature in a curve or surface where the geometry changes in a particular way. A "cusp neighborhood," therefore, would typically refer to a local neighborhood around such a cusp point. A cusp is characterized by having a point where the curve (or manifold) has a sharp point or a change in direction that cannot be smoothed out.

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