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 $f:X\to Y$ is continuous if $x\approx x^{\prime }⟹f(x)\approx f(x^{\prime })$. Intuitively, infinitely close points go to infinitely close points.

$f$ is bounded if $x\sim x^{\prime }⟹f(x)\sim f(x^{\prime })$ 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”.