Difference between revisions of "Real analysis"
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'''Definition''': The ''lower Darboux integral'' <math>L(f)</math> of <math>f</math> over <math>[a,b]</math> is given by <cmath>U(f) = \sup\{L(f,P) \mid P \text{ is a partition of } [a,b]\}.</cmath> | '''Definition''': The ''lower Darboux integral'' <math>L(f)</math> of <math>f</math> over <math>[a,b]</math> is given by <cmath>U(f) = \sup\{L(f,P) \mid P \text{ is a partition of } [a,b]\}.</cmath> | ||
− | '''Definition''': A function is ''integrable'' if <math>U(f) = L(f)</math>. We identify this common value as the ''Darboux integral'' of <math>f</math> over <math>[a,b]</math> and is denoted by <cmath>\int_a^b f(x)\,dx = U(f) = L(f).</cmath> | + | '''Definition''': A function is ''Darboux integrable'' if <math>U(f) = L(f)</math>. We identify this common value as the ''Darboux integral'' of <math>f</math> over <math>[a,b]</math> and is denoted by <cmath>\int_a^b f(x)\,dx = U(f) = L(f).</cmath> |
+ | |||
+ | Various shortcomings of Darboux (and, consequently, Riemann) integration exist. A famous example is the [[Direchlet function]] which is not Darboux-integrable on any intervale of <math>\mathbb{R}</math>. Furthermore, the [[monotone convergence theorem]] is false in the context of Darboux integrals which motivates the [[Lebesgue integral]]. | ||
== See also == | == See also == |
Revision as of 01:45, 5 November 2022
Broadly speaking, real analysis is the study of the real numbers and its topological properties, sequences and series of real numbers, and properties of real-valued functions. Some properties that are studied in the real numbers are the construction of the real numbers, convergence of sequences, subsets of the plane as metric spaces, limits, notions of continuity, differentiation, and integration.
A common description of real analysis courses is that real analysis is the formal rigorous study of single-variable calculus with proofs. This view does have merit to it because most (if not all) of the theorems typically presented to students in courses in single-variable calculus are proven rigorously; however, one should note that courses in real analysis also spend considerable amount of time on pathological examples with little concern for applications, and one also aims to generalize and prove results rather than apply results to calculate numerical answers to exercises as one typically does in calculus.
Contents
Topics
Construction of the real numbers
The entirety of real analysis is built upon the real numbers, particularly with the notion of completeness in mind. Intuitively, this is described as the fact that the real numbers lack the existence of any "holes" unlike the rational numbers
(for instance, the set
has no smallest element in the rational numbers). This property of the real numbers is known as the least upper bound property.
Two particularly known constructions of the real numbers are via Cauchy sequences and Dedekind cuts, both of which take and construct
as a completion of
.
Sequences of real numbers
A sequence is a function . Conventionally, sequences are typically denoted by the notation
where
is denoted by
. In the case where
, we can denote
by
.
In real analysis, particular attention is paid attention to the convergence and divergence of sequences. Intuitively, the idea of convergence is captured by the notion that the sequence "approaches" some value as becomes arbitrarily large. Also important is the notion of Cauchy sequences which intuitively describe sequences whose terms become arbitrarily close to each other as
becomes arbitrarily large.
Limits
A large problem with the intuitive notion of a sequence converging to some value is that "approaching" is not only vague, but is also handwavy and lacks mathematical precision. Limits solve this problem by precisely defining the notion of convergence of a sequence.
Definition: Let be a sequence of real numbers. The sequence
converges to the limit
provided that for every
, there exists
such that for every
, we have
If
converges to
, then we say that
or
as
.
This definition can be shown to be equivalent to the likely more familiar definition of a limit of a function.
Definition: Let . The limit of the function
as
approaches
is
provided that for every
, there exists a
such that that for every
and
, we have
. Notationally, we say that
or
as
.
Limits are a key tool in the definition of continuity, derivatives, and any result in real analysis that relies upon sequences.
Continuity
A common analogy used in calculus classes for continuity is a function whose graph can be drawn without lifting up one's pencil--that is, the graph has no breaks or jumps. While intuitive, it turns out that this notion of continuity is actually very misleading, in fact, a continuous function may have discontinuities at points not in its domain (for example, is continuous at all points in its domain yet is "visually discontinuous" at
). This calls for a more precise notion of continuity.
Definition: Let . The function
is continuous at
provided that for every sequence
in
converging to
, we have
as
. In other words,
preserves convergence.
Definition: The function is said to be continuous on
if it is continuous at every point in
. The function
Definition: The function is said to be continuous if it is continuous at every point in
.
This definition of continuity is equivalent to the more familiar definition of continuity from calculus below.
Definition. Let . The function
is continuous at
provided that for every
, there exists a
such that for every
and
, we have
.
Continuity can also be generalized to topological spaces and described in terms of preimages of open sets. Furthermore, various other notions of continuity such as Lipschitz continuity, uniform continuity, and absolute continuity are studied in real analysis.
Differentiation
Derivatives are central notion in calculus and various fields such as the sciences and economics where derivatives can be interpreted as the instantaneous rate of change at a point.
Definition: The function is differentiable at
if the limit
exists.
Definition: Let . The function
is differentiable on
provided that it is differentiable at every point in
.
In the context of real analysis, differentiability is a condition which guarantees continuity, but not the other way around. In particular, there exist functions that are continuous on but differentiable nowhere (for instance, Weierstrass's function). One can also define higher-order derivatives by inductively differentiating the derivative of a function.
Integration
In calculus, integrals are introduced as the approximation of the area under the curve of a continuous functions by rectangles as the number of rectangles becomes arbitrarily large. This construction is widely used in many fields of knowledge including math itself and is known as a Riemann integral. In real analysis, Riemann integrals of non-continuous functions are also considered and the notion of Lebesgue integration is also developed.
In some texts, the notion of Darboux integrals are introduced first which are then noted as special cases of Riemann sums.
Let be bounded.
Definition: A partition of
is a finite set such that
.
Definition: The upper Darboux sum of
with respect to
is given by
Definition: The lower Darboux sum of
with respect to
is given by
Definition: The upper Darboux integral of
over
is given by
Definition: The lower Darboux integral of
over
is given by
Definition: A function is Darboux integrable if . We identify this common value as the Darboux integral of
over
and is denoted by
Various shortcomings of Darboux (and, consequently, Riemann) integration exist. A famous example is the Direchlet function which is not Darboux-integrable on any intervale of . Furthermore, the monotone convergence theorem is false in the context of Darboux integrals which motivates the Lebesgue integral.
See also
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