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.
Differentiation and regularity Differentiation measures the local rate of change of a function. In one variable, the derivative gives the slope of the best linear approximation to a function near a point. This point of view extends to several variables, where differentiability is expressed in terms of approximation by a linear map.
In addition to proving rigorously the fundamental properties of the derivative from calculus, such as the chain rule, real analysis also establishes the main theorems about the derivative, such as the mean value theorem and some of its generalizations like the Cauchy mean value theorem. Roughly speaking, the mean value theorem relates the derivative of a function to its average rate of change over intervals. The mean value theorem is important because it leads to an effective way to estimate the error in certain approximations or averaging processes.
Real analysis studies not only existence of derivatives, but also degrees of regularity. A function may be continuous but nowhere differentiable, differentiable but not continuously differentiable, or smooth (having derivatives of all orders) but not analytic (equal to its Taylor series). One theorem that is sometimes shown at the elementary level is Darboux's theorem, that the derivative of a differentiable function satisfies the intermediate value property; weaker than continuity, but still a restricted class of functions. In more advanced analysis, regularity questions become central. The classical derivative turns out to be very inconvenient in many applications, such as to differential equations where many problems have a more natural variational formulation using integration rather than differentiation. The functions one must consider are often not differentiable, and so generalizations of differentiability, and even what constitutes a function, are considered. Regularity then involves questions about whether the generalized derivatives are honest derivatives, and how smooth solutions to equations are. This is a common theme in measure theory, partial differential equations, Sobolev spaces, and the calculus of variations.
Integration and measure Integration in analysis gives rigorous meaning to averaging, accumulation, and area. The Riemann integral formalizes the integral using approximations by finite sums over intervals. The Riemann integral is put on a firm foundation in basic real analysis courses. The fundamental theorem of calculus, which relates integration and differentiation, is proved for the Riemann integral. The fundamental theorem underlies many exact computations in elementary integral calculus by reducing definite integrals to antiderivatives. The mean value theorem for integration states that the average value of a continuous function over an interval, defined by a Riemann integral, is equal to the value of the function at some point inside the interval. The mean value theorem is important in deriving analytic estimates in analytic applications; for example, one form of the Taylor remainder uses it.
The Lebesgue integral is introduced in more advanced real analysis. Instead of approximating the integral by finite sums using partitions of the domain, the Lebesgue integral partitions the range instead. While the idea is similar to the Riemann integral, it differs from the Riemann integral in that one must formulate the area of horizontal slabs that are not connected rectangles, but may be distributed over complicated sets in the domain. Thus formulating the Lebesgue integral leads naturally to measure theory, the fundamental question of which is how to measure subsets of the real line – that is, assign a notion of length – if the subsets are allowed to be very complicated.
One reason for the success of the Lebesgue integral in analysis is that it is more compatible with limiting processes. For example, under relatively mild hypotheses, the limit of the Lebesgue integrals of members of a sequence of functions that converges only pointwise is equal to the integral of the pointwise limit (dominated convergence theorem). Moreover, the Lebesgue integral admits a wider class of functions as integrable. The Riemann integral, although not widely used in mathematical analysis, still has its uses however: most numerical integration relies on the Riemann rather than Lebesgue approach to integration.
Measure theory treats length, area, volume, mass, and probability as instances of a general concept of measure. The theory allows one to identify functions that are equal except on a set of measure zero, and one says that such functions are equal almost everywhere. In probability theory, the relevant notion is almost surely. An example is tossing a fair coin an infinite sequence of times. It is possible that the coin could come up tails every time, but this event has zero probability: it can be essentially excluded from any probabilistic argument.
The monotone convergence theorem, Fatou's lemma, dominated convergence theorem, and Fubini's theorem are basic theorems about the Lebesgue integral. Measure-theoretic real analysis also supplies the language of probability and of many function spaces. In probability theory, expected values are integrals with respect to probability measures. In the theory of Lp spaces, functions are studied according to integrability properties and are identified when they agree almost everywhere.
Series, function sequences, and representation
Infinite series are one of the basic limiting processes of real analysis. A numerical series is defined by the convergence of its partial sums, and questions about its absolute convergence, conditional convergence, and rearrangements illustrate the difference between finite and infinite additions. Power series and Taylor series connect infinite series with differentiablity and analyticity.
Sequences and series are used to approximate and represent functions. Uniform convergence gives conditions under which continuity, integrability, and sometimes differentiability, pass to the limit function. Power series can be used to represent some functions locally, while Fourier series represent periodic functions in terms of trigonometric series. These lead from elementary analysis to questions of approximation theory (how well a function is represented by a series or partial sum), harmonic analysis (what can be inferred about the regularity of the function from its representations), and the theory of function spaces (what classes of functions can be represented, and how well, in a certain way).
Metric spaces and function spaces