Cusp

A pointed singularity on a curve or boundary where smoothness fails because two tangent directions coalesce into a single tangent with higher contact; the curve is not regular there and curvature typically blows up.

A cuspidal singularity (cusp) is a point where a single branch of a curve has a self-tangency producing a pointed, nontransverse local geometry; algebraically the quadratic part is a square and higher-order terms produce a single tangent direction with multiplicity.