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An example of a constant function is y(x) = 4, because the value of y(x) is 4 regardless of the input value x. 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.
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
The signum function restricted to the domain {} is locally constant.. In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.
[3] A function whose value remains unchanged (i.e., a constant function). [4] Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question. For example, a general quadratic function is commonly written as: + +,
Given such a constant k, the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by {(,): =}. A direct proportionality can also be viewed as a linear equation in two variables with a y -intercept of 0 and a slope of k > 0, which corresponds to linear growth .
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.
For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set.
Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x.. If there exists a positive number r > 0 such that f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x.