Limit of a function at a finite value

[(477, 493)]

A function is an assignment of inputs to outputs so that each input has exactly one output associated to it.

Specifying a function $f: X \to Y$ is to specify the following:

1. A set $X$, called the domain of $f$.
2. A set $Y$, called the codomain of $f$. (Some authors call this the range, but other authors make a distinction between codomain and range.)
3. For each $x \in X$¹, a single element of $Y$, which we write as $f(x)$.

In particular, given any subset $S \subset X$, there is a function $f|_S$ called the restriction of $f$ to $S$ given by $f|_S(s) = f(s)$ for all $s \in S$. It therefore makes sense to say that $f$ is defined on $S$.

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1. https://hyunjongkimmath.github.io/mathbook/notation.basic.html#$\in$↩