Heine-Cantor Theorem

The Heine-Cantor Theorem states that if a space is compact—meaning it is closed and bounded—then any continuous function defined on this space must be uniformly continuous. In simple terms, this means that for such spaces, you can find a single rule that guarantees the function's behavior won't have abrupt changes: once you know how close inputs are, you can confidently predict how close outputs will be, uniformly across the whole space. This property ensures stable, predictable behavior of the function throughout the entire compact set.