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# Topic: Continuous function (topology)

###### In the News (Sat 20 Jul 19)

 math lessons - Continuous function In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper. All polynomials are continuous, and so are the exponential functions, logarithms, square root function and trigonometric functions (on their respective domains). www.mathdaily.com /lessons/Continuous_function   (1072 words)

 Springer Online Reference Works The latter property is sometimes taken as the starting point for the definition of a generalized function; together with the theorem on the completeness of the space of generalized functions it leads to an equivalent definition of generalized functions [8]. is a derivative of a continuous function in Usually the support of a function (or distribution) is defined as the closure of the set of points where it is non-zero. eom.springer.de /g/g043810.htm   (1132 words)

 topology. The Columbia Encyclopedia, Sixth Edition. 2001-05 Topology is sometimes referred to popularly as “rubber-sheet geometry” because a figure can be changed to that of an equivalent figure by bending, stretching, twisting, and the like, but not by tearing or cutting. Topology is concerned with those properties of geometric figures that are invariant under continuous transformations. A continuous transformation, also called a topological transformation or homeomorphism, is a one-to-one correspondence between the points of one figure and the points of another figure such that points that are arbitrarily close on one figure are transformed into points that are also arbitrarily close on the other figure. www.bartleby.com /65/to/topology.html   (892 words)

 Topology glossary A function from one space to another is continuous if the preimage of every open set is open. Two sets A and B in a space are functionally separated if there is a continuous function from the space into the interval [0,1] with the property that A is mapped to 0 and B is mapped to 1. A partition of unity is a set of continuous functions from a space to [0,1] such that any point has a neighbourhood where all but a finite number are identically zero, and the sum of all them at every point is 1. www.ebroadcast.com.au /lookup/encyclopedia/lo/Local_base.html   (1004 words)

 What Is Topology? Topology is a relatively new branch of mathematics; most of the work has been done since 1900. Algebraic topology often uses the combinatorial structure of a space to calculate the various groups associated to that space. Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex variables. www.math.uwaterloo.ca /PM_Dept/What_Is/Topology/topology.shtml   (470 words)

 Continuous function - Article from FactBug.org - the fast Wikipedia mirror site In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper. All polynomials are continuous, and so are the exponential functions, logarithms, square root function and trigonometric functions (on their respective domains). www.factbug.org /cgi-bin/a.cgi?a=6122   (1122 words)

 Distribution They extend the concept of derivative to all continuous functions and beyond and are used to formulate generalized solutions of partial differential equations. R is a smooth (= infinitely often differentiable) function which is identically zero except on some bounded set, then ∫fφdx is a real number which linearly and continuously depends on φ. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution. www.ebroadcast.com.au /lookup/encyclopedia/di/Distribution.html   (1409 words)

 [No title] Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html). An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together. Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold. www.ornl.gov /sci/ortep/topology/defs.txt   (5717 words)

 PlanetMath: continuous This is version 7 of continuous, born on 2001-10-21, modified 2006-10-22. Object id is 439, canonical name is Continuous. I agree completely with you that the non-standard definition is not shorter or simpler once you take into account the fact that, to use it you first have to define non-standard numbers. planetmath.org /encyclopedia/Continuous.html   (303 words)

 ipedia.com: Continuous function Article   (Site not responding. Last check: 2007-10-21) All polynomials are continuous, and so are the exponential functions, logarithms, square root function and trigonometric functions. For example, if a child undergoes continuous growth from 1m to 1.5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have equalled 1.25m. As a consequence, if f(x) is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c, f(c) must equal zero. www.ipedia.com /continuous_function.html   (1143 words)

 PlanetMath: another proof of the non-existence of a continuous function that switches the rational and the irrational ... PlanetMath: another proof of the non-existence of a continuous function that switches the rational and the irrational numbers "another proof of the non-existence of a continuous function that switches the rational and the irrational numbers" is owned by neapol1s. This is version 4 of another proof of the non-existence of a continuous function that switches the rational and the irrational numbers, born on 2006-11-12, modified 2006-11-13. planetmath.org /encyclopedia/ProofOfTheNonExistenceOfAContinuousFunctionThatSwitchesTheRationalAndTheIrrationalNumbers.html   (168 words)

 Topology Topology is a branch of pure mathematics that deals with the abstract relationships found in geometry and analysis. The study of topology requires a solid background in calculus and a facility with logic and proofs. Some have described topology as seeking the most general kind of space in which continuity can be defined. www.stetson.edu /~mhale/topology/index.htm   (462 words)

 Continuous function (topology) - Wikipedia, the free encyclopedia In topology and related areas of mathematics a continuous function is a morphism between topological spaces. A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. Dually, for a function f from a set S to a topological space, one defines the initial topology on S by letting the open sets of S be those subsets A of S for which f(A) is open in X. en.wikipedia.org /wiki/Continuity_(topology)   (1258 words)

 Note on functions over streams. A function over streams is a function whose domain and codomain are both streams, where by a stream is meant a countably infinite sequence. For example, the function that copies the input through, except that lowercase characters are replaced by their uppercase equivalent is a stream processing function. A continuous function is however something more than a function on streams: to produce the next character of output, it should be necessary to examine at most a finite prefix of the input. homepages.inf.ed.ac.uk /v1phanc1/datastream.html   (767 words)

 Topology The first, continuous topology, centers on the effects of compactness and metrization, is represented here by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces. The second, geometric topology, focuses on the connectivity properties of topological spaces and provides the core results from general topology that serve as background for subsequent courses in geometry and algebraic topology. In classical topology, this relation is simple and clear: "An open set is a neighborhood of a point if and only if this point belongs to this open set." In early period of fuzzy topology, "membership relation" was similarly defined. www.wordtrade.com /science/mathematics/topology.htm   (2132 words)

 A Treatise on Class Theory   (Site not responding. Last check: 2007-10-21) The class topology of f(x) = 2x is not Hausdorff, however, and in fact consists of two circles and a point, and the only open set containing the point is the whole space. Moreover, in a continuous function, classes seem to come in groups of classes that look the same and are "near" each other, it seems to me that investigation of classification of these groups of classes would be beneficial. The problem of finding continuous roots seems to be related to solving the original problem given certain restrictions of the classes to be paired and the relation of different pairs based on the class topology of the original function. members.tripod.com /~SeraphSama/class3.html   (507 words)

 Topology A function f:X \to Y is called a _continuous map_ between topological spaces (X,t) and (Y,s) if for every V in s, f^-1(V) is in t (i.e., the inverse image of an open set must be open). This is a generalization of the notion of continuous function given in calculus as we now show. The reader should now check that continuity in the sense of calculus of a function from R to R is equivalent to continuity as a map of topological spaces, with respect to the topology m. www.georgetown.edu /faculty/kainen/topology.html   (1132 words)

 A Function-Based Data Model for Visualization The base space is analogous to the independent variables of a function while fiber space is analogous to the dependent variables of a function, and fiber bundle section is analogous to a function (field). These functions are continuous in nature, but sampled or discretized in a fashion often dictated by the specific computations to be performed. Operations imply a process of transformation between different functions of this class, whether it is solved as a set of partial differential equations that define flow of heat or generating pixels as a rendering of some geometry. www.research.ibm.com /people/l/lloydt/dm/function/dm_fn.htm   (4765 words)

 Topology Reading Course Students are expected to become familiar with the basic concepts and methodology of point-set topology: separation properties, connectedness, and compactness, as well as subspaces, quotient spaces, and the properties of continuous mappings. Show that if X is the set of all differentiable functions defined on (0,1) with (f,g) in R provided that f' = g', then R is an equivalence relation on X. More generally, show that defining two functions to be related if their difference is a constant function is an equivalence relation. That is, excluding the trivial smallest topology (consisting of only the empty set and the entire set, called the co-discrete'' topology) and the trivial largest topology (which consists of _all_ subsets and is called the discrete'' topology), there are 27 nontrivial topologies. www.georgetown.edu /faculty/kainen/topol-02.html   (1223 words)

 Algebraic Topology: Topology A map or continuous function from a topological space (X,OX) to a topological space (Y,OY) is a function from X to Y such that the preimage of any member of OY is a member of OX. In order to check that a given function is continuous, it suffices to check that the inverse images of the members of a subbasis for the open sets are open again. The topology on A defined by F is the weakest topology (i.e., the smallest collection OA) for which all these functions become continuous. www.win.tue.nl /~aeb/at/algtop-2.html   (1509 words)

 Topological Preliminaries Topology is one of (quite a few) mathematical theories that permeate other branches of Mathematics connecting them into one coherent whole. Most of the examples will be drawn on the 2-dimensional plane but, given the definitions of the distance and neighborhood could be carried over to the 1- and many dimensional cases. A continuous function f that has a continuous inverse function is called a topological transformation. www.cut-the-knot.org /do_you_know/topology.shtml   (759 words)

 Map (mathematics) Summary A mapping is a function that is represented by two sets of objects with arrows drawn between them to show the relationships between the objects. An onto function has a relationship in which every object in the codomain is paired with at least one object in the domain. In many branches of mathematics, the term denotes a function with a property specific to that branch, such as a continuous function in topology, a linear transformation in linear algebra, etc. www.bookrags.com /Map_(mathematics)   (877 words)

 Compact Open Topology This brings in the function that maps both a and b to 1, and that is not part of g∩h. If f is a continuous function from x into y, and k is compact in x, the functions that are within ε of f, on the domain k, form an open set. It is more intuitive to say, "The functions close to f on k", rather than "The finite intersection of functions that map various compact sets k into various open sets u." Of course x and y could be exotic spaces, whence the compact open topology cannot be expressed in a simpler, equivalent form. www.mathreference.com /top-cs,cot.html   (940 words)

 Topology history A second way in which topology developed was through the generalisation of the ideas of convergence. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated. Fréchet continued the development of functional by defining the derivative of a functional in 1904. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Topology_in_mathematics.html   (1456 words)

 Topology MAT 530 This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous. A set A of functions is said to be equicontinuous at a point x if for every positive epsilon there is a neighborhood U of x such that f(U) lies in the epsilon-neighborhood of f(x) for all f from A (i.e., uniformly on f) The Arzela-Ascoli theorem is very useful in analysis. The Urysohn lemma states that for a normal topological space X and two disjoint closed subsets A and B of it, there exists a continuous function from X to [0,1] that is 0 on A and 1 on B. www.math.sunysb.edu /~timorin/mat530.html   (2896 words)

 PlanetMath: discontinuous may be continuous (continuous in all points in can be made into an analytic function on the whole complex plane. Cross-references: mapping, topological spaces, signum function, complex plane, analytic function, real number, continuous, modification, clear, closure, one-sided limits, properties, points, boundary, continuous at, function, interval, open set planetmath.org /encyclopedia/Discontinuous.html   (234 words)

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