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For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y. [2]
with domain, the range of , sometimes denoted or (), [4] may refer to the codomain or target set (i.e., the set into which all of the output of is constrained to fall), or to (), the image of the domain of under (i.e., the subset of consisting of all actual outputs of ). The image of a function is always a subset of the codomain of the ...
The image of a function is the image of its entire domain, also known as the range of the function. [3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of f . {\displaystyle f.}
A function f is continuous at a point which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that | () | < for all such that (,) <. In other words, φ may be chosen small enough for having the image by f of the interval of radius φ centered at a {\displaystyle a} contained in the ...
This is one of the reasons for which, in mathematical analysis, "a function from X to Y " may refer to a function having a proper subset of X as a domain. [note 2] For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers, typically a subset ...
A domain of a real smooth function can be the real coordinate space (which yields a real multivariable function), a topological vector space, [1] an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above in § Algebraic structure and are subspaces of the space of continuous ...
The natural logarithm function ln : (0, +∞) → R is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the exponential function, if defined with the set of real numbers as the domain and the codomain, is not surjective (as its range is the set of positive real numbers).
As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c. For example, the function y(x) = 4 is the specific constant function where the output value is c = 4. The domain of this function is the set of all real numbers. The image of this function is the singleton set {4}.