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Approximately Continuous
In mathematics, particularly in mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the limit of a function, ordinary limit with an approximate limit. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory. Definition Let E \subseteq \mathbb^n be a Lebesgue measurable set, f\colon E \to \mathbb^k be a measurable function, and x_0 \in E be a point where the Lebesgue density of E is 1. The function f is said to be ''approximately continuous'' at x_0 if and only if the approximate limit of f at x_0 exists and equals f(x_0). Properties A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain. The concept of approximate continuity can be extended beyond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
