IMITS OF

Consider a function f : D → Y, and suppose x0 ∈ D′, the derived set of D. Since there exist points x ∈ D arbitrarily close to x0, we can consider the values f (x) as x gets closer and closer to x0. In particular, if we let x get arbitrarily close to x0, does f (x) get arbitrarily close to an element in Y? In order to consider such a limit as well defined, we require that f (x) get arbitrarily close to the same element in Y no matter how we let x approach x0. We formalize these ideas in defining the limit of a function, a concept that is crucial to the development of analysis.