differentiability

Ecco una proposta per l’excerpt, descrittiva e in stile più manualistico, senza “this guide”:

Discontinuity of a real function is the failure of continuity at a point, occurring when the limit does not exist, is not finite, or does not coincide with the function value. The main types include removable, jump, and infinite discontinuities, each reflecting a distinct way in which local behavior may break down.

Piecewise functions are functions defined by different formulas on different parts of their domain, with each formula applying under specified conditions.