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For a function to have an inverse, it must be one-to-one.If a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function = defined on the whole of is not one-to-one since = for any .
For example, the restriction of any function (even one as interesting as the Dirichlet function) to any subset on which it is constant will be continuous, although this fact is as uninteresting as constant functions.
For example, the cosine function is injective when restricted to the interval [0, π]. The image of this restriction is the interval [−1, 1], and thus the restriction has an inverse function from [−1, 1] to [0, π], which is called arccosine and is denoted arccos. Function restriction may also be used for "gluing" functions together.
In mathematics, a corestriction [1] of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.
A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space.
For example, if X is a closed subvariety of an affine variety Y and f is the inclusion, then f # is the restriction of regular functions on Y to X. See #Examples below for more examples. Regular functions
For example, the map : [,) defined by the polynomial () = is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset is dense in ; with additional effort (using the inverse function theorem for instance), it can be shown that = {}, which confirms that this set is indeed dense in .
One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} and has domain and codomain both equal to the real numbers .