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In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain . Real-valued functions of a real variable (commonly called real functions ) and real-valued functions of several real variables are the main object of study of calculus and ...
A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity ...
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.
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.
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.)
Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots.
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In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. [1]