Difference between revisions of "Real analysis"

Revision as of 22:57, 4 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.

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 $\mathbb R$ lack the existence of any "holes" unlike the rational numbers $\mathbb Q$ (for instance, the set $\{x \in \mathbb{Q} \mid x^2 < 2\}$ 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 $\mathbb{Q}$ and construct $\mathbb{R}$ as a completion of $\mathbb{Q}$ .

Sequences of real numbers

A sequence is a function $f:\mathbb{N}\to\mathbb{R}$ . Conventionally, sequences are typically denoted by the notation $(s_n)_{n = k}^{\infty} = (s_k,s_{k + 1},\ldots)$ where $f(n)$ is denoted by $s_n$ . In the case where $k = 1$ , we can denote $(s_n)_{n = k}^{\infty}$ by $(s_n)_{n \in \mathbb{N}}$ .

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 $n$ becomes arbitrarily large. Also important is the notion of Cauchy sequences which intuitively describe sequences whose terms become arbitrarily close to each other as $n$ 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 $(s_n)$ be a sequence of real numbers. The sequence $(s_n)$ converges to the limit $s\in\mathbb{R}$ provided that for every $\varepsilon > 0$ , there exists $N\in\mathbb{N}$ such that for every $n \ge N$ , we have $|s_n - s| < \varepsilon.$ If $(s_n)$ converges to $s$ , then we say that $\lim_{n\to\infty}s_n = s$ or $s_n \rightarrow s$ as $n \to \infty$ .

This definition can be shown to be equivalent to the likely more familiar $\varepsilon-\delta$ definition of a limit of a function.

Definition: Let $S\subseteq\mathbb{R}$ . The limit of the function $f:S\to\mathbb{R}$ as $x$ approaches $x_0$ is $L$ provided that for every $\varepsilon > 0$ , there exists a $\delta > 0$ such that that for every $x \in S$ and $0<|x-x_0|<\delta$ , we have $|f(x)-L|<\varepsilon$ . Notationally, we say that $\lim_{x\to x_0}f(x)=L$ or $f(x) \rightarrow L$ as $x \rightarrow x_0$ .

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, $f(x) = 1/x$ is continuous at all points in its domain yet is "visually discontinuous" at $x = 0$ ). This calls for a more precise notion of continuity.

Definition: Let $S\subseteq\mathbb{R}$ . The function $f:S\to\mathbb R$ is continuous at $x_0\in S$ provided that for every sequence $(x_n)$ in $S$ converging to $x_0$ , we have $f(x_n)\rightarrow f(x_0)$ as $x_n\rightarrow x_0$ . In other words, $f$ preserves convergence.

Definition: The function $f$ is said to be continuous on $T\subseteq S$ if it is continuous at every point in $T$ . The function

Definition: The function $f$ is said to be continuous if it is continuous at every point in $S$ .

This definition of continuity is equivalent to the more familiar definition of continuity from calculus below.

Definition. Let $S\subseteq\mathbb{R}$ . The function $f:S\to\mathbb R$ is continuous at $x_0\in S$ provided that for every $\varepsilon > 0$ , there exists a $\delta > 0$ such that for every $x \in S$ and $|x - x_0|<\varepsilon$ , we have $|f(x)-f(x_0)|<\varepsilon$ .

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.

@@ Line 31: / Line 31: @@
 === 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, <math>f(x) = 1/x</math> is continuous at all points in it's domain yet is "visually discontinuous" at <math>x = 0</math>). This calls for a more precise notion of 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, <math>f(x) = 1/x</math> is continuous at all points in its domain yet is "visually discontinuous" at <math>x = 0</math>). This calls for a more precise notion of continuity.
 '''Definition''': Let <math>S\subseteq\mathbb{R}</math>. The function <math>f:S\to\mathbb R</math> is continuous at <math>x_0\in S</math> provided that for every sequence <math>(x_n)</math> in <math>S</math> converging to <math>x_0</math>, we have <math>f(x_n)\rightarrow f(x_0)</math> as <math>x_n\rightarrow x_0</math>. In other words, <math>f</math> preserves convergence.

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