Construction of the real numbers The theorems of real analysis rely on the properties of the (established)
real number system. The real number system consists of an
uncountable set (\mathbb{R}), together with two
binary operations denoted and \cdot, and a
total order denoted . The operations make the real numbers a
field, and, along with the order, an
ordered field. The real number system is the unique
complete ordered field, in the sense that any other complete ordered field is
isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \mathbb{Q}) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the
least upper bound property (see below).
Order properties of the real numbers The real numbers have various
lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an
ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is
total, and the real numbers have the
least upper bound property:
Every nonempty subset of \mathbb{R} that has an upper bound has a least upper bound that is also a real number. These
order-theoretic properties lead to a number of fundamental results in real analysis, such as the
monotone convergence theorem, the
intermediate value theorem and the
mean value theorem. However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in
functional analysis and
operator theory generalize properties of the real numbers – such generalizations include the theories of
Riesz spaces and
positive operators. Also, mathematicians consider
real and
imaginary parts of complex sequences, or by
pointwise evaluation of
operator sequences.
Topological properties of the real numbers Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a
topological space, the real numbers has a
standard topology, which is the
order topology induced by order . Alternatively, by defining the
metric or
distance function d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{\geq 0} using the
absolute value function as the real numbers become the prototypical example of a
metric space. The topology induced by metric d turns out to be identical to the standard topology induced by order . Theorems like the
intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \mathbb{R} only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.
Sequences A
sequence is a
function whose
domain is a
countable,
totally ordered set. The domain is usually taken to be the
natural numbers, although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. Of interest in real analysis, a
real-valued sequence, here indexed by the natural numbers, is a map a : \N \to \R : n \mapsto a_n. Each a(n) = a_n is referred to as a
term (or, less commonly, an
element) of the sequence. A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses: (a_n) = (a_n)_{n \in \N}=(a_1, a_2, a_3, \dots) . A sequence that tends to a
limit (i.e., \lim_{n \to \infty} a_n exists) is said to be
convergent; otherwise it is
divergent. (
See the section on limits and convergence for details.) A real-valued sequence (a_n) is
bounded if there exists M\in\R such that |a_n| for all n\in\mathbb{N}. A real-valued sequence (a_n) is
monotonically increasing or
decreasing if a_1 \leq a_2 \leq a_3 \leq \cdots or a_1 \geq a_2 \geq a_3 \geq \cdots holds, respectively. If either holds, the sequence is said to be
monotonic. The monotonicity is
strict if the chained inequalities still hold with \leq or \geq replaced by . Given a sequence (a_n), another sequence (b_k) is a
subsequence of (a_n) if b_k=a_{n_k} for all positive integers k and (n_k) is a strictly increasing sequence of natural numbers.
Limits and convergence Roughly speaking, a
limit is the value that a
function or a
sequence "approaches" as the input or index approaches some value. (This value can include the symbols \pm\infty when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to
calculus (and
mathematical analysis in general) and its formal definition is used in turn to define notions like
continuity,
derivatives, and
integrals. (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.) The concept of limit was informally introduced for functions by
Newton and
Leibniz, at the end of the 17th century, for building
infinitesimal calculus. For sequences, the concept was introduced by
Cauchy, and made rigorous, at the end of the 19th century by
Bolzano and
Weierstrass, who gave the modern
ε-δ definition, which follows.
Definition. Let f be a real-valued function defined on {{nowrap|E\subset\mathbb{R}.}} We say that
f(x) tends to L as x approaches x_0, or that
the limit of f(x) as x approaches x_0 is L if, for any \varepsilon>0, there exists \delta>0 such that for all x\in E, 0 implies that |f(x) - L| . We write this symbolically as f(x)\to L\ \ \text{as}\ \ x\to x_0 , or as \lim_{x\to x_0} f(x) = L . Intuitively, this definition can be thought of in the following way: We say that f(x)\to L as x\to x_0, when, given any positive number \varepsilon, no matter how small, we can always find a \delta, such that we can guarantee that f(x) and L are less than \varepsilon apart, as long as x (in the domain of f) is a real number that is less than \delta away from x_0 but distinct from x_0. The purpose of the last stipulation, which corresponds to the condition 0 in the definition, is to ensure that \lim_{x \to x_0} f(x)=L does not imply anything about the value of f(x_0) itself. Actually, x_0 does not even need to be in the domain of f in order for \lim_{x \to x_0} f(x) to exist. In a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large.
Definition. Let (a_n) be a real-valued sequence. We say that (a_n)
converges to a if, for any \varepsilon > 0, there exists a natural number N such that n\geq N implies that |a-a_n| . We write this symbolically as a_n \to a\ \ \text{as}\ \ n \to \infty ,or as\lim_{n \to \infty} a_n = a ; if (a_n) fails to converge, we say that (a_n)
diverges. Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the
limit of f(x) as x increases without bound, notated \lim_{x \to \infty} f(x). Reversing the inequality x\geq M to x \leq M gives the corresponding definition of the limit of f(x) as x
decreases without bound, {{nowrap|\lim_{x \to -\infty} f(x).}} Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant. In these cases, the concept of a
Cauchy sequence is useful.
Definition. Let (a_n) be a real-valued sequence. We say that (a_n) is a
Cauchy sequence if, for any \varepsilon > 0, there exists a natural number N such that m,n\geq N implies that |a_m-a_n| . It can be shown that a real-valued sequence is Cauchy
if and only if it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, (\R, |\cdot|), is a
complete metric space. In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.
Uniform and pointwise convergence for sequences of functions In addition to sequences of numbers, one may also speak of
sequences of functions on E\subset \mathbb{R}, that is, infinite, ordered families of functions f_n:E\to\mathbb{R}, denoted (f_n)_{n=1}^\infty, and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known as
pointwise convergence and
uniform convergence, that need to be distinguished. Roughly speaking, pointwise convergence of functions f_n to a limiting function f:E\to\mathbb{R}, denoted f_n \rightarrow f, simply means that given any x\in E, f_n(x)\to f(x) as n\to\infty. In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.
Uniform convergence requires members of the family of functions, f_n, to fall within some error \varepsilon > 0 of f for
every value of x\in E, whenever n\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted f_n\rightrightarrows f, such a value of N must exist for any \varepsilon>0 given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough N, the functions f_N, f_{N+1}, f_{N+2},\ldots are all confined within a 'tube' of width 2\varepsilon about f (that is, between f - \varepsilon and f+\varepsilon)
for every value in their domain E. The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. For example, a sequence of continuous functions (see
below) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.
Karl Weierstrass is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.
Compactness Compactness is a concept from
general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being
closed and
bounded. (In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a
closed set contains all of its
boundary points, while a set is
bounded if there exists a real number such that the distance between any two points of the set is less than that number. In \mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points,
closed intervals, and their finite unions. However, this list is not exhaustive; for instance, the set \{1/n:n\in\mathbb{N}\}\cup \{0}\ is a compact set; the
Cantor ternary set \mathcal{C}\subset [0,1] is another example of a compact set. On the other hand, the set \{1/n:n\in\mathbb{N}\} is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set [0,\infty) is also not compact because it is closed but not bounded. For subsets of the real numbers, there are several equivalent definitions of compactness.
Definition. A set E\subset\mathbb{R} is compact if it is closed and bounded. This definition also holds for Euclidean space of any finite dimension, \mathbb{R}^n, but it is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the
Heine-Borel theorem. A more general definition that applies to all metric spaces uses the notion of a subsequence (see above).
Definition. A set E in a metric space is compact if every sequence in E has a convergent subsequence. This particular property is known as
subsequential compactness. In \mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above. Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general. The most general definition of compactness relies on the notion of
open covers and
subcovers, which is applicable to topological spaces (and thus to metric spaces and \mathbb{R} as special cases). In brief, a collection of open sets U_{\alpha} is said to be an
open cover of set X if the union of these sets is a superset of X. This open cover is said to have a
finite subcover if a finite subcollection of the U_{\alpha} could be found that also covers X.
Definition. A set X in a topological space is compact if every open cover of X has a finite subcover. Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact.
Continuity A
function from the set of
real numbers to the real numbers can be represented by a
graph in the
Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken
curve with no "holes" or "jumps". There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to be
equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the first definition given below, f:I\to\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain. Some possibilities include I=\R, the whole set of real numbers, an
open interval I = (a, b) = \{x \in \R \mid a or a
closed interval I = [a, b] = \{x \in \R \mid a \leq x \leq b\}. Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.
Definition. If I\subset \mathbb{R} is a non-degenerate interval, we say that f:I \to \R is
continuous at p\in I if \lim_{x \to p} f(x) = f(p). We say that f is a
continuous map if f is continuous at every p\in I. In contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of \lim_{x\to p} f(x), must also hold in order for f to be continuous at p:
(i) f must be defined at p, i.e., p is in the domain of f;
and (ii) f(x)\to f(p) as x\to p. The definition above actually applies to any domain E that does not contain an
isolated point, or equivalently, E where every p\in E is a
limit point of E. A more general definition applying to f:X\to\mathbb{R} with a general domain X\subset \mathbb{R} is the following:
Definition. If X is an arbitrary subset of \mathbb{R}, we say that f:X\to\mathbb{R} is
continuous at p\in X if, for any \varepsilon>0, there exists \delta>0 such that for all x\in X, |x-p| implies that |f(x)-f(p)| . We say that f is a
continuous map if f is continuous at every p\in X. A consequence of this definition is that f is
trivially continuous at any isolated point p\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between
topological spaces (which includes
metric spaces and \mathbb{R} in particular as special cases). This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.
Definition. If X and Y are topological spaces, we say that f:X\to Y is
continuous at p\in X if f^{-1} (V) is a
neighborhood of p in X for every neighborhood V of f(p) in Y. We say that f is a
continuous map if f^{-1}(U) is open in X for every U open in Y. (Here, f^{-1}(S) refers to the
preimage of S\subset Y under f.)
Uniform continuity Definition. If X is a subset of the
real numbers, we say a function f:X\to\mathbb{R} is
uniformly continuous on X if, for any \varepsilon > 0, there exists a \delta>0 such that for all x,y\in X, |x-y| implies that |f(x)-f(y)| . Explicitly, when a function is uniformly continuous on X, the choice of \delta needed to fulfill the definition must work for
all of X for a given \varepsilon. In contrast, when a function is continuous at every point p\in X (or said to be continuous on X), the choice of \delta may depend on both \varepsilon
and p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless. On a compact set, it is easily shown that all continuous functions are uniformly continuous. If E is a bounded noncompact subset of \mathbb{R}, then there exists f:E\to\mathbb{R} that is continuous but not uniformly continuous. As a simple example, consider f:(0,1)\to\mathbb{R} defined by f(x)=1/x. By choosing points close to 0, we can always make |f(x)-f(y)| > \varepsilon for any single choice of \delta>0, for a given \varepsilon > 0.
Absolute continuity Definition. Let I\subset\mathbb{R} be an
interval on the
real line. A function f:I \to \mathbb{R} is said to be
absolutely continuous on I if for every positive number \varepsilon, there is a positive number \delta such that whenever a finite sequence of
pairwise disjoint sub-intervals (x_1, y_1), (x_2,y_2),\ldots, (x_n,y_n) of I satisfies :\sum_{k=1}^{n} (y_k - x_k) then :\sum_{k=1}^{n} | f(y_k) - f(x_k) | Absolutely continuous functions are continuous: consider the case
n = 1 in this definition. The collection of all absolutely continuous functions on
I is denoted AC(
I). Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.
Differentiation The notion of the
derivative of a function or
differentiability originates from the concept of approximating a function near a given point using the "best" linear approximation. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a. A function f:\mathbb{R}\to\mathbb{R} is
differentiable at a if the
limit :f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} exists. This limit is known as the
derivative of f at a, and the function f', possibly defined on only a subset of \mathbb{R}, is the
derivative (or
derivative function)
of f. If the derivative exists everywhere, the function is said to be
differentiable. As a simple consequence of the definition, f is continuous at
a if it is differentiable there. Differentiability is therefore a stronger regularity condition than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see
Weierstrass's nowhere differentiable continuous function). It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on. One can classify functions by their
differentiability class. The class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions. The class C^1 consists of all
differentiable functions whose derivative is continuous; such functions are called
continuously differentiable. Thus, a C^1 function is exactly a function whose derivative exists and is of class C^0. In general, the classes
C^k can be defined
recursively by declaring C^0 to be the set of all continuous functions and declaring
C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}. In particular,
C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict. Class C^\infty is the intersection of the sets
C^k as
k varies over the non-negative integers, and the members of this class are known as the
smooth functions. Class C^\omega consists of all
analytic functions, and is strictly contained in C^\infty (see
bump function for a smooth function that is not analytic).
Series A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers. The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers. The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms. Instead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound. The series is assigned the value of this limit, if it exists. Given an (infinite)
sequence (a_n), we can define an associated
series as the formal mathematical object {{nowrap|a_1 + a_2 + a_3 + \cdots = \sum_{n=1}^{\infty} a_n,}} sometimes simply written as \sum a_n. The
partial sums of a series \sum a_n are the numbers s_n=\sum_{j=1}^n a_j. A series \sum a_n is said to be
convergent if the sequence consisting of its partial sums, (s_n), is convergent; otherwise it is
divergent. The
sum of a convergent series is defined as the number {{nowrap|s = \lim_{n \to \infty} s_n.}} The word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms. For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the
Riemann rearrangement theorem for further discussion). An example of a convergent series is a
geometric series which forms the basis of one of Zeno's famous
paradoxes: :\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1 . In contrast, the
harmonic series has been known since the Middle Ages to be a divergent series: :\sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty . (Here, "=\infty" is merely a notational convention to indicate that the partial sums of the series grow without bound.) A series \sum a_n is said to
converge absolutely if \sum |a_n| is convergent. A convergent series \sum a_n for which \sum |a_n| diverges is said to
converge non-absolutely. It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a series that converges non-absolutely is :\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2 .
Taylor series The Taylor series of a
real or
complex-valued function ƒ(
x) that is
infinitely differentiable at a
real or
complex number a is the
power series :f(a) + \frac{f'(a)}{1!} (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \frac{f^{(3)}(a)}{3!} (x-a)^3 + \cdots. which can be written in the more compact
sigma notation as : \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} where
n! denotes the
factorial of
n and
ƒ (
n)(
a) denotes the
nth
derivative of
ƒ evaluated at the point
a. The derivative of order zero
ƒ is defined to be
ƒ itself and and 0! are both defined to be 1. In the case that , the series is also called a Maclaurin series. A Taylor series of
f about point
a may diverge, converge at only the point
a, converge for all
x such that |x-a| (the largest such
R for which convergence is guaranteed is called the
radius of convergence), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at that point. If the Taylor series at a point has a nonzero
radius of convergence, and sums to the function in the
disc of convergence, then the function is
analytic. The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that the
exponential function, the
logarithm, the
trigonometric functions and their
inverses are extended to functions of a complex variable.
Fourier series for a
square wave. Fourier series are an important tool in real analysis. Fourier series decomposes
periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely
sines and cosines (or
complex exponentials). The study of Fourier series typically occurs and is handled within the branch
mathematics >
mathematical analysis >
Fourier analysis.
Integration Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface. The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the
method of exhaustion. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed. By considering approximations consisting of a larger and larger ("infinite") number of smaller and smaller ("infinitesimal") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value. The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices ("refinements") are considered. Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind. Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a "measure" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist "non-measurable" subsets for which an area cannot be assigned.
Riemann integration The Riemann integral is defined in terms of
Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a
closed interval of the real line; then a
tagged partition \cal{P} of [a,b] is a finite sequence : a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\! This partitions the interval [a,b] into n sub-intervals [x_{i-1},x_i] indexed by i=1,\ldots, n, each of which is "tagged" with a distinguished point t_i\in[x_{i-1},x_i]. For a function f bounded on [a,b], we define the
Riemann sum of f with respect to tagged partition \cal{P} as :\sum_{i=1}^{n} f(t_i) \Delta_i, where \Delta_i=x_i-x_{i-1} is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The
mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, \|\Delta_i\| = \max_{i=1,\ldots, n}\Delta_i. We say that the
Riemann integral of f on [a,b] is S if for any \varepsilon>0 there exists \delta>0 such that, for any tagged partition \cal{P} with mesh \| \Delta_i \| , we have ::\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| This is sometimes denoted \mathcal{R}\int_{a}^b f=S. When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower)
Darboux sum. A function is
Darboux integrable if the upper and lower
Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal. In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former. The
fundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense.
Lebesgue integration and measure Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the
domains on which these functions can be defined. The concept of a
measure, an abstraction of length, area, or volume, is central to Lebesgue integral
probability theory.
Distributions Distributions (or
generalized functions) are objects that generalize
functions. Distributions make it possible to
differentiate functions whose derivatives do not exist in the classical sense. In particular, any
locally integrable function has a distributional derivative.
Relation to complex analysis Real analysis is an area of
analysis that studies concepts such as sequences and their limits, continuity,
differentiation,
integration and sequences of functions. By definition, real analysis focuses on the
real numbers, often including positive and negative
infinity to form the
extended real line. 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, expressibility as
power series, and satisfying the
Cauchy integral formula. 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. However, results such as the
fundamental theorem of algebra are simpler when expressed in terms of complex numbers. Techniques from the
theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by
residue calculus. ==Important results==