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The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f. For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. [1] [2] In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval [t, t 1] when started at the time-t state variable x(t)=x. [3]
A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x) is such a complex valued function, it may be decomposed as f(x) = g(x) + ih(x), where g and h are real-valued functions. In other words ...
The set X is called the domain of the function [2] and the set Y is called the codomain of the function. [3] Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Paul Milgrom and Ilya Segal (2002) observe that the traditional envelope formula holds for optimization problems with arbitrary choice sets at any differentiability point of the value function, [5] provided that the objective function is differentiable in the parameter:
The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values. [1] [2] For example, if the function f is defined by f (x) = 2 x 2 – 3 x + 1, then assigning the value 3 to its argument x yields the function value 10, since f (3) = 2·3 2 – 3·3 + 1 = 10.
More precisely, whereas a function satisfying an appropriate summability condition defines an element of L p space, in the opposite direction for any f ∈ L p (X) and x ∈ X which is not an atom, the value f(x) is undefined. Though, real-valued L p spaces still have some of the structure described above in § Algebraic structure.
A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All step functions are simple.