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Let X and Y be two metric spaces, and F a family of functions from X to Y.We shall denote by d the respective metrics of these spaces.. The family F is equicontinuous at a point x 0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x 0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x 0, x) < δ.
Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F. More formally, let X and Y be topological spaces. The set of continuous functions : has a natural topology called the compact-open topology. A normal family is a pre-compact ...
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.
Other families of functions obtained by picking one arbitrary function from each T(X) → T(X) would not have such uniformity. They are called "ad hoc polymorphic functions". Parametricity is the abstract property enjoyed by the uniformly acting families such as twice, which distinguishes them from ad hoc families.
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters. [1] Common examples are parametrized (families of) functions, probability distributions, curves, shapes, etc. [citation needed]
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...
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The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed for R but should generalize to any language and library.