f: X → Y is continuous if and only if f (x) limn→∞ f(xn) for every sequence xn → x.
Care must be taken to distinguish between continuous and open mappings. A function f: X → Y is continuous if f-1(T) is open in X whenever T is open in Y. It is called an open mapping if f (S) is open in Y whenever S is open in X. An open mapping preserves open sets. If an open mapping has inverse, then the inverse is continuous (exercise 2.69). In general, continuous functions are not open mappings (example 2.78). However, every continuous function on an compact domain is an open mapping (exercise 2.76), as is every bounded linear function (proposition 3.2).