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In mathematics, a multivalued function, [1] multiple-valued function, [2] many-valued function, [3] or multifunction, [4] is a function that has two or more values in its range for at least one point in its domain. [5] It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued ...
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
The implicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows. [4] Let ϕ(x 1, x 2, …, x n) be a continuous function with continuous first order partial derivatives, and let ϕ evaluated at a point (a, b) = (a 1, a 2, …, a n, b) be zero:
It has filter,mapandreduce functions; list comprehensions, dictionaries, sets, and generator expressions. [78] The standard library has two modules (itertools and functools) that implement functional tools borrowed from Haskell and Standard ML. [79] Its core philosophy is summarized in the Zen of Python (PEP 20), which includes aphorisms such ...
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the tent map is topologically conjugate to the logistic map. As a special case, taking f(x) = x + 1, one has the iteration of g(x) = h −1 (h(x) + 1) as g n (x) = h −1 (h(x) + n), for any function h.
A higher-order function is a function that takes a function as an argument or returns one as a result. This is commonly used to customize the behavior of a generically defined function, often a looping construct or recursion scheme. Anonymous functions are a convenient way to specify such function arguments. The following examples are in Python 3.