History of calculus | Infinity | Mathematics of infinitesimals | Nonstandard analysis

Law of continuity

The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler used the law of continuity to calculate the area of the circle by representing it as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. The transfer principle provides a mathematical implementation of the law of continuity in the context of the hyperreal numbers. A related law of continuity concerning intersection numbers in geometry was promoted by Jean-Victor Poncelet in his "Traité des propriétés projectives des figures". (Wikipedia).

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Continuity: definition of (actually sequential continuity); examples; standard definition involving neighborhoods; examples.

From playlist Course 6: Introduction to Analysis

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From playlist Calculus 1 Exam 1 Playlist

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From playlist Differential Equations

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Continuity using Limits

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From playlist Continuity Using Limits

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From playlist Calculus I

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From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)

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From playlist Limits

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Continuity of functions and different types of discontinuities, and the relationship between continuity and differentialbility.

From playlist Calculus Chapter 2: Limits (Complete chapter)

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From playlist Limits and Continuity

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From playlist Calculus 1

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From playlist Numerical Analysis and Scientific Computing

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From playlist Continuity Using Limits

Related pages

Infinitesimal | Pierre Varignon | Transcendental law of homogeneity | Gottfried Wilhelm Leibniz | Hyperreal number | Geometry | Intersection number | Transfer principle