Continuous vs Bounded

Nonstandard analysis gives a nice link between continuous and bounded functions, which are small and large-scale notions.¹

Let \(X\) and \(Y\) be topological spaces. Identify them with their nonstandard extensions. A function \(f: X \rightarrow Y\) is continuous if \(x \approx x' \implies f(x) \approx f(x')\). Intuitively, infinitely close points go to infinitely close points.

\(f\) is bounded if \(x \sim x' \implies f(x) \sim f(x')\) where \(\sim\) means “is a limited distance from”. Intuitively, points that are a limited distance from each other go to points a limited distance from each other.

Notice these definitions are almost the same. Only the equivalence relation of “being infinitely close” is replaced with the equivalence relation of “being a limited distance away”.