Introduction to :

Continuous functions as we've introduced them here are just the tip of the iceberg. The field of point-set topology defines more general notions of open and closed sets and then defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function. We can have much weirder functions that don't have numbers as inputs or outputs, and yet are still continuous.