Discontinuous Collocation Method

picture info	Discontinuous Collocation Method Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values. The oscillation of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * in an essential discontinuity, oscillation measures the failure of a limit to exist; the limit is constant. A special case is if the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Continuous Function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' 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 intuitive notions of continuity, and considered only continuous functions. The 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 and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	One-sided Limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches a "from the right" or "from above") can be denoted: \lim_f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x+) The limit as x increases in value approaching a (x approaches a "from the left" or "from below") can be denoted: \lim_f(x) \quad \text \quad \lim_\, f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x-) If the limit of f(x) as x approaches a exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit \lim_ f(x) does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a "two-sided limit". It is possible for exactly one of the two one-sided limits to exist (while t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Regulated Function In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book "Livre IV: Fonctions d'une variable réelle". Definition Let ''X'' be a Banach space with norm , , - , , ''X''. A function ''f'' : , ''T''→ ''X'' is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true: * for every ''t'' in the interval , ''T'' both the left and right limits ''f''(''t''−) and ''f''(''t''+) exist in ''X'' (apart from, obviously, ''f''(0−) and ''f''(''T''+)); * there exists a sequence of step functions ''φ''''n'' : , ''T''→ ''X'' converging uniformly to ''f'' (i.e. with respect to the supremum norm , , - , , ∞). It requires a little work to show that these two conditions are equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Riemann Integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration. Overview Let be a non-negative real-valued function on the interval , and let be the region of the plane under the graph of the function and above the interval . See the figure on the top right. This region can be expressed in set-builder notation as S = \left \. We are interested in measuring the area of . Once we have measured it, we will denote the area in the usual way by \int_a^b f(x)\,dx. The basic idea of the Riemann integral is to use very simple approximations for the area of . By taking better ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]