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In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value () of some function. An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition.
The pointwise limit of continuous functions does not have to be continuous: the continuous functions (marked in green) converge pointwise to a discontinuous function (marked in red). Suppose that X {\displaystyle X} is a set and Y {\displaystyle Y} is a topological space , such as the real or complex numbers or a metric space , for example.
For example, every ordered group is residuated, and the division defined by the above coincides with notion of division in a group. A less trivial example is the set Mat n (B) of square matrices over a boolean algebra B, where the matrices are ordered pointwise. The pointwise order endows Mat n (B) with pointwise meets, joins and complements.
This implies that for any function f of any Hölder class α > 0, the Fourier series converges everywhere to f(x). It is also known that for any periodic function of bounded variation, the Fourier series converges. In general, the most common criteria for pointwise convergence of a periodic function f are as follows:
Examples of almost sure convergence; Example 1; Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be quite certain that one day the number will become zero, and will stay zero forever after. Example 2
For example, Fejér's theorem shows that if one replaces ordinary summation by Cesàro summation then the Fourier series of any continuous function converges uniformly to the function. Further, it is easy to show that the Fourier series of any L 2 function converges to it in L 2 norm.
The Dirichlet function is an archetypal example of the Blumberg theorem. The Dirichlet function can be constructed as the double pointwise limit of a sequence of ...
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).