Real analysis is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
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Real analysis is an area of analysis, which studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. However, the scope of real analysis is restricted to the real numbers, and this defines the range of tools available.
Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressability as power series, and satisfying the Cauchy integral formula.
However, in real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. Also results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers.
On the other hand, the real numbers have several important analytic properties of their own. They are totally ordered, and have the least upper bound property, and these properties lead to a number of important results in real analysis, such as the monotone convergence theorem, the intermediate value theorem and the mean value theorem.
However, whilst results in real analysis are generally stated for real numbers, they may still be used in other areas of mathematics - such as by considering real and imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. Conversely, techniques from other areas are often used in real analysis - such as evaluation of real integrals by residue calculus.
The foundation of real analysis is the construction of the real numbers from the rational numbers, usually either by Dedekind cuts, or by completion of Cauchy sequences.
Key concepts in real analysis are real sequences and their limits, continuity, differentiation, and integration.
Real analysis is also used as a starting point for other areas of analysis, such as complex analysis, functional analysis, and harmonic analysis, as well as motivating the development of topology, and as a tool in other areas, such as applied mathematics.
Important results include the Bolzano-Weierstrass and Heine-Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.