A cusp is a point or edge that marks the end of one stage and the beginning of another. It's often seen in nature.
Imagine looking at the shape of a leaf: it has a sharp edge, called a cusp, where the stem meets the blade of the leaf. The cusp of a tooth is the pointed part where the teeth meet. If you're talking about the economy, a recession and growth might be considered two stages, with an economic cusp marking the transition between them.
How common is "cusp"?
Word cusp is considered rare in modern English. It has a balanced usage among all categories: speech, fiction, newspapers and academic texts.
Definitions
noun
A sharp point or pointed end.
An important moment when a decision is made that will determine future events.
A point of a curve where the curve is continuous but has no derivative, but such that it has a derivative at every nearby point.
A point made by the intersection of two curved lines or curved structures, a common motif in Gothic architecture.
A boundary between zodiacal signs and houses.
Any of the pointed parts of a canine tooth or molar.