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## Real analysisReal analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions.The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction. Then the real numbers are either introduced axiomatically, or they are constructed from sequences of rational numbers. Initial consequences are derived, most importantly the properties of the absolute value such as the triangle inequality and the Bernoulli inequality.
The concept of The concept of continuity may now be defined via limits, one can show that the sum, product, composition and quotient of continuous functions is continuous, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem. Then one can do integration (Riemann and Lebesgue) and prove the Fundamental Theorem of Calculus, typically using the mean value theorem. At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated. We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.
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