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Def/Extension of functions
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Definition of Extension of a function: Suppose that $X$ and $Y$ are sets, and $S$ is a subset of $X$. Suppose that $f \colon S \rightarrow Y$ is a function. An extension of $f$ to $X$ is a function $\tilde f \colon X \rightarrow Y$ such that: $$\forall s \in S, \tilde f(s) = f(s).$$ Under this condition, we say that $\tilde f$ extends $f$.
Logical Connections
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The following statements and definitions logically rely on the material of this page: Def/Permutation, and State/Injective functions have left inverses
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