Limits and Continuity

Theorem

Letbe a function with domainand suppose thatis a limit point ofso that every open neighbourhood ofcontains infinitely many points of

the following statements are equivalent:

  1. f is continuous at

Proof

Suppose thatis continuous atIfis any sequence inwiththen it follows that forwe can findsuch thatimplies

Hence

Conversely suppose thatIf is any sequence inwithandis the subsequence ofconsisting of those terms different fromthensinceand

The remaining subsequenceofsatisfiesfor allsosoandis continuous at