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From Intelligent Perception

Suppose we have sets $X$ and $Y$, a function $f: X → Y$, and a subset $A$ of $X$. Then the restriction function $f|_A: A → Y$ of $f$ is given by $$f|_A(x) = f(x)$$ for all $x∈A$.

In that case, the original function $f$ is an extension of $f|_A$.