Continuous Map
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Continuous Map
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value of the function. This means that there is no abrupt changes in value, known as Classification of discontinuities, discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on Intuition, intuitive notions of continuity, and considered only continuous functions. The (ε, δ)-definition of limit, epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real number, ...
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Mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (calculus and mathematical analysis, analysis). There is no general consensus about its exact scope or epistemology, epistemological status. Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of mathematical object, abstract objects. These objects are either abstractions from nature (such as natural numbers or "a line (mathematics), line"), or (in modern mathematics) abstract entities that are defined by their basic properties, called axioms. A proof (mathematics), proof consists of a succession of applications of some inference rule, deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are consider ...
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