LYCOS RETRIEVER 
Continuous Function: Set
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In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by $$ W(x) = \sum_{k=0}^{\infty} a^k\cos(b^k\pi x)\text{,} $$ where $a$ is a real number with $0 1 + 3\pi/2$. This example was first published by du Bois-Reymond in 1875.
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A continuous function is more complicated than a momentary function. It is event driven: it must respond to user interaction through the mouse, keyboard, and user defined devices. As a result, continuous functions do not actually take over control of StudioTools when they execute. Rather, they define a context under which UI events are interpreted. StudioTools reads these events and transfers them to a group of event handlers defined by the continuous function. Thus a continuous function is a set of five callback functions: init, mouse down, mouse move, mouse up and cleanup.
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The set of points of the unit circle at which a continuous complex-valued function in the open unit disk has limits along curves (asymptotic values) is of type F(sd) and, in general, has no other properties. The author shows that for continuous complex-valued functions defined in a cube, this set of "curvilinear convergence" does not even need to be a Borel set. He asks whether such an example can be given for real-valued functions.
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Real-valued functions of a real variable which are continuous with respect to the density topology on both the domain and the range are called density continuous. A typical continuous function is nowhere density continuous. The same is true of a typical homeomorphism of the real line. A subset of the real line is the set of points of discontinuity of a density continuous function if and only if it is a nowhere dense F\sigma set. The corresponding characterization for the approximately continuous functions is a first category F\sigma set. An alternative proof of that result is given.
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The author generalizes the result of McMillan (1966) to the effect that the set of curvilinear convergence of a continuous function f from D into Z is of type F(sd). The generalization considers f as a continuous function from D into a compact metric space E. Topologizing the set of closed sets C(E) of E with the Hausdorff metric and letting E be any closed set in C(E), it is shown that the set of all x ( C such that there is a boundary path v at x with the cluster set of f along v contained in some set of E is of type F(sd). Taking E to be the set of all singletons {y}, y ( E (which is closed in C(Z)) McMillan's theorem is obtained.
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A probability density function f of a continuous distribution is not unique. Note that the values of f on a finite (or even countable) set of points could be changed to other nonnegative values, and properties (a), (b), and (c) would still hold. The critical fact is that only integrals of f are important.
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