11 Limits and Continuity in Metric Spaces
The concept of "limit" can be extended to metric spaces in a natural way, just by using distance in place of absolute value in the usual definitions. We can define both the limit of a sequence and the limit of a function in this way:
But we can rephrase these definitions in terms of open balls to be more geometrically intuitive. This works more effectively for the definition of continuity instead of the limit of a function:
We then get a very nice characterization of continuity in terms of limits of sequences:
We define $f$ to be continuous if it is continuous at every point of its domain, $A.$ We then see that $f$ is continuous if and only if for every convergent sequence $\{x_n\}_{n=1}^\infty,$ $ \ds\lim_{n\to\infty}f(x_n) = f\big(\lim_{n\to\infty} x_n\big).$
We get another, somewhat surprising, characterization of continuity purely in terms of open sets. This shows that continuity is actually a topological rather than a metric property.