Real analysis
| <math>\frac{d}{dx} \sin x=?\,</math> | This article/section deals with mathematical concepts appropriate for late high school or early college. |
Real analysis is a field in mathematics that focuses on the set of real numbers, their properties, sequences and functions. Included in this branch of mathematics are the concepts of limits and convergence, calculus, and properties of real-valued functions such as continuity. It also includes measure theory.
For the purposes of this article, "analysis" will be limited to the generalization and extension of the concepts of calculus, using the concepts of elementary point-set topology.
The reader should be quite familiar with the concepts of calculus, especially limits and continuity. In particular, the reader should be comfortable with the ramifications of the phrase "for every epsilon".
Open sets
Open sets (and, by extension, closed sets, which are just the complements of open sets) are the fundamental concept of analysis. Analysis and topology are really just the study of open sets.
Before giving the definition of open sets in Euclidean space, we present some examples. Readers who are aware of the general intuitive notion of open sets should find these examples familiar.
In one dimension
The simplest open sets in 1-dimensional Euclidean space (formally called <math>\mathbb{R}^1</math>; informally called the real numbers of the "real line") are open intervals. An open interval consists of those numbers lying strictly between two endpoints a and b. In set-theoretic notation:
- <math>\{ x\ |\ a < x < b \}\,</math>
A shorter notation for this set consists of the two endpoints in parentheses:
- <math>( a, b )\,</math>
A closed interval (we will have more to say about closed sets later) would include the endpoints. It is commonly denoted with brackets:
- <math>[ a, b ] = \{ x\ |\ a \le x \le b \}</math>
An interval that includes one endpoint but not the other is called semi-open:
- <math>[ a, b ) = \{ x\ |\ a \le x < b \}</math>
- <math>( c, d ] = \{ x\ |\ c < x \le d \}</math>
When drawing pictures of intervals, those same symbols are typically used:
- Need a picture here!
Open intervals are not the only open sets. Any union of open intervals is an open set. For example:
- <math>\bigcup_{N \textrm{\ is\ an\ integer\ } \geq 2} (N,N+1/N)</math>
Bizarrely defined sets like the one above are commonly used as examples and counterexamples in analysis and topology.
In two or more dimensions
In two or more dimensions the situation becomes more complicated, because even simple open sets can come in an endless variety of shapes. The fundamental open set (equivalent to an open interval) is the open neighborhood, also called an open ball. An open neighborhood has a center point and a nonzero radius, and is the set of all points whose distance from the center is strictly less than that radius. In set-theoretic notation:
- <math>\{ x\ |\ \|x-C\| < r \}\,</math>
The double-stroke absolute value sign is the norm or the metric distance function. In the common case it is the Euclidean/Pythagorean distance:
- <math>\|a-b\| = \sqrt{(a_1-b_1)^2 + (a_2-b_2)^2}\,</math> in two dimensions (similarly for higher dimensions)
The double-stroke absolute value sign is similar to the usual absolute value operation, generalized to arbitrary dimensions or other metric spaces.
It is easy to see that, in two dimensions, an open neighborhood is the interior of a circle. It does not include the actual boundary of the circle, because it consists of the points whose distance from <math>C</math> is strictly less than <math>r</math>. This point is crucialâthe whole subject of analysis and topology depends on it!
To draw a picture of an open neighborhood, use a circle bounded by a dotted line:
- Need a picture here!
(To make a closed ball, the formula would be:
- <math>\{ x\ |\ \|x-C\| \le r \}\,</math>
and the picture would be a solid circle. But open neighborhoods are the important sets from a theoretical standpoint.)