Continuity space | Wikipedia audio article

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00:01:27 1 History
00:03:37 2 Real functions
00:03:46 2.1 Definition
00:05:07 2.1.1 Definition in terms of limits of functions
00:08:21 2.1.2 Definition in terms of neighborhoods
00:09:34 2.1.3 Definition in terms of limits of sequences
00:11:24 2.1.4 Weierstrass and Jordan definitions (epsilon–delta) of continuous functions
00:13:05 2.1.5 Definition in terms of control of the remainder
00:14:43 2.1.6 Definition using oscillation
00:18:41 2.1.7 Definition using the hyperreals
00:20:20 2.2 Construction of continuous functions
00:21:33 2.3 Examples of discontinuous functions
00:27:25 2.4 Properties
00:30:42 2.4.1 Intermediate value theorem
00:34:23 2.4.2 Extreme value theorem
00:34:31 2.4.3 Relation to differentiability and integrability
00:35:28 2.4.4 Pointwise and uniform limits
00:36:09 2.5 Directional and semi-continuity
00:36:19 3 Continuous functions between metric spaces
00:38:28 3.1 Uniform, Hölder and Lipschitz continuity
00:39:57 4 Continuous functions between topological spaces
00:41:18 4.1 Continuity at a point
00:42:23 4.2 Alternative definitions
00:43:47 4.2.1 Sequences and nets
00:45:14 4.2.2 Closure operator definition
00:47:51 4.3 Properties
00:49:42 4.4 Homeomorphisms
00:50:01 4.5 Defining topologies via continuous functions
00:52:00 5 Related notions
00:54:45 6 See also
00:56:19 7 Notes
00:57:09 8 References
00:59:29 Related notions
01:01:29 See also



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SUMMARY
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps at each point in time when money is deposited or withdrawn, so the function M(t) is discontinuous.