📂Functions

Expansion and Contraction of a Function

Definition¹

Let’s assume that function $f : X \to Y$ is given. Let’s also assume that $U \subset X \subset V$ holds.

Contraction Mapping

We call $f |_{U} \to Y$ a contraction mapping of $f$ if it satisfies the following.

$$ f|_{U} : U \to Y \quad \text{and} \quad f|_{U}(x) = f (x),\quad \forall x \in U $$

Extension

We call $\tilde{f} \to Y$ an extension of $f$ if it satisfies the following.

$$ \tilde{f} : V \to Y \quad \text{and} \quad \tilde{f}(x) = f (x),\quad \forall x \in X $$

Explanation

Usually, instead of the translated terms contraction mapping (also known as restriction) and extension, the direct English pronunciations [restriction] and [extension] are used.

Simply put, it’s about narrowing or widening the domain of the function while keeping its shape unchanged.

According to the definition, it’s obvious that $f$ is a restriction of $\tilde{f}$, and an extension of $f|_{U}$.