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More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}
The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
This indicates the importance of the domain of discourse, which specifies which values n can take. [note 1] In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a logical conditional. For example,
1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc. This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages, this is immediately a problem, since syntax rules are expected to generate finite words.
When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value .
Usually "domain" means "domain of definition", that is, the set of values for which the function is defined. However, in some contexts, such as complex analysis "domain" refers to a restricted domain. The restricted domain is a subset of the domain of definition. It can be chosen arbitrarily.
The value for must be solved for in this manner because for all values of , is only defined for < < +, and is periodic (with period ). This means that the inverse function will only give values in the domain of the function, but restricted to a single period.
In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point. A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p. There is another type of limit of a function, namely the ...