Proof That a Function on a First Countable Space is Continuous at a Point if and Only if it is Sequentially Continuous at The Point

Theorem

A functiondefined on a first countable spaceis continuous at a pointif and only if it is sequentially continuous at the point.

Proof

Letbe a function on a setnd letbe a nested local base at

Supposeis not continuous. Then an open setexists such thatand for every

Thus for everyanexists such thatso

Hencebut