Obélix
The analyzes real is the branch of the analyzes which studies the real Ensemble S of and the functions of real variables. She studies concepts like the limiting continuations and their , the Continuité, the derivation, the integration and the continuations of functions.
Concepts
The presentation of the real analysis in the advanced works usually starts with simple Démonstration S of results of the naive Théorie of the units, a clear definition of the concept of function, an introduction to the natural whole and the important demonstration of the Raisonnement by recurrence.
Then, the real numbers or are introduced by axioms, or are built starting from continuations of rational numbers. Important consequences result some, such as for example the properties of the absolute Value, the triangular Inégalité or the Inégalité of Bernoulli.
Convergence
The concept of convergence , exchange with the analysis, is introduced via the limiting of continuations. Several laws which control the process of calculation of limit can be established, and various limits can be calculated. The series S, which are particular continuations, are also studied in analysis. The whole series are used to define several important functions properly, such as the exponential function and the goniometrical functions. Various important types of subsets of real numbers, such as the open units, the closed units, the compact units and their properties are introduced.
Continuity
The concept of Continuité can then be defined starting from the limits, the properties of algebra of the limits result some; the sum, the product, the made up one and the quotient of continuous functions are continuous, and the important theorem of the intermediate values is shown.
Derivation and integration
The concept of Dérivée can be defined like a limit of a rate of variation, and the rules of calculation of a derivative can be shown rigorously. A central theorem is the Théorème of the finished increases.
We can also make integration (of Riemann and Lebesgue) and show the fundamental Théorème of the analysis, by using the theorem of the average.
More advanced concepts
At this point, it is useful to study the concepts of continuity and convergence within a more abstract framework, with an aim of considering spaces of functions. This is treated in topological spaces or metric spaces. The concepts of compactness, Complétude, of connexity, Uniform continuity, Separability, applications lipschitziennes, contracting applications are defined and studied.
We can consider limit functions and try to change the order of the integrals, derivative and limits. The uniform concept of Convergence is important in this context. For that, it is useful to have rudimentary knowledge on the produced normalized vector spaces and scalar. Finally the Taylor series can also be approached.
See too
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Analyze
- Analyze complexes
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