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In mathematics, given a function f: A → B, the set A is called the domain, or domain of definition of f.

A well-defined function must map every element of the domain to an element of its codomain. So, for example, the function:

f: x → 1/x

has no valid value for f(0). It is thus not a function on the set R of real numbers; R can't be its domain. It is usually either defined as a function on R \ {0}, or the "gap" is plugged by specifically defining f(0); for example:

f: x → 1/x , x ≠ 0

f: 0 → 0

The domain of given function can be restricted to a subset. Suppose that g: A → B, and S ⊆ A. Then the restriction of g to S is written:

g|_S: S → B