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An elderly Tibetan woman holding a prayer wheel demonstrates the continuity theory. Despite their age, older adults generally maintain the same traditions and beliefs. The continuity theory of normal aging states that older adults will usually maintain the same activities, behaviors, relationships as they did in their earlier years of life. [1]
A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity. As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
In summary, this previous sentence's statement of absolute continuity is false. The contiguity property replaces this requirement with an asymptotic one: Q n is contiguous with respect to P n if the "limiting support" of Q n is a subset of the limiting support of P n. By the aforementioned logic, this statement is also false.
Continuity (mathematics), the opposing concept to discreteness; common examples include Continuous probability distribution or random variable in probability and statistics Continuous game , a generalization of games used in game theory
Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions whose intersection is not lower hemicontinuous.
Continuity in probability is a sometimes used as one of the defining property for Lévy process. [1] Any process that is continuous in probability and has independent increments has a version that is càdlàg. [2] As a result, some authors immediately define Lévy process as being càdlàg and having independent increments. [3]