Real analysis is the part of mathematical analysis, especially as taught in undergraduate and graduate courses, that develops calculus rigorously over the real numbers and Euclidean spaces. Introductory real analysis is sometimes called advanced calculus, and studies limits, continuity, compactness, differentiation, integration, and series. More advanced courses often include measure theory, Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real variable, in contrast to the theory of complex variables.
Scope
Real numbers The real numbers are the basic setting of real analysis, which begins with their construction. The real numbers are distinguished from the rational numbers by their completeness. Roughly speaking, the real numbers have no gaps. This completeness can be formalized in several equivalent ways, one of which is the least upper bound property. This states that if a non-empty set of real numbers is bounded above, meaning that all of its elements are less than some number (an upper bound), then there is a least upper bound, that is an upper bound that is smaller than all of the others. Most of the theorems that are proved in real analysis rely on completeness in one way or another. Some examples where its relevance are most apparent are as follows. The convergence of bounded monotone sequences, that is sequences that are increasing (or decreasing), is essentially equivalent to the least upper bound property, stated in sequence form. Completeness is also reflected in the intermediate value theorem: the continuous image of an interval is again an interval, so continuous functions cannot create gaps.
Limits and convergence The concept of a limit underlies many of the ideas of calculus, such as the derivative. It is therefore of fundamental importance in real analysis, which provides the rigorous justification for calculus. Limits describe how a sequence, function, or family of functions behave under a limiting process, such as letting an index tend to infinity, or a letting point become very large or approach another point. The formal language of limits defines continuity, differentiation, integration, infinite series, and various kinds of approximations and asymptotics. A recurring problem in real analysis is not just whether a limit exists, but how well one object approximates another. A convergent sequence approximates its limit, or a differentiable function is approximated by a linear function coming from the derivative. But real analysis can provide not only tools to justify the existence of these limits and how to calculate them, but also quantitative estimates of how good the approximation is. One example of this is the Taylor remainder, which gives an effective and computable constant that determines how well the linear approximation (or higher-order Taylor polynomial) approximates a function on an interval. For sequences of functions, real analysis distinguishes between different modes of convergence. A sequence of functions converges pointwise if it converges at every point, but, roughly speaking, the rate of convergence may vary from point to point. It converges uniformly if it converges at all points at a comparable rate. Uniform convergence can be understood intuitively in terms of the graphs of the functions in the sequence: it means that, for any given thin error band around the limiting function, all but finitely many functions in the sequence stay within the band. Pointwise convergence means that this is true for an error band around each point, but the finite set of functions which must be excluded varies from point to point. For sequences of functions, pointwise convergence often fails to preserve operations on the limit function. For example, it is not generally true that the pointwise limit of a sequence of continuous functions is continuous, or that the integral of the functions in a sequence passes to the integral of the limit function. But the uniform limit of continuous functions is continuous, and one can exchange integration and uniform limits on suitable domains. Uniform convergence is therefore important for many applications of real analysis. Questions such as "when is differentiation under the integral allowed?" or "when can I integrate an infinite sum term-by-term?" are typical examples in which uniform convergence provides a simple answer.
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