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:

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