Continuity: Regions

IN THE BLUE CORNER, Advanced Calculus:

"

Multiple Integrals  

§ 1. Introduction 

In this chapter, we shall discuss double and triple integrals.  We shall follow as closely as possible the analogy with the theory of simple integrals developed in the previous chapter. 

1.1 REGIONS
     We have already discussed in Chapter 1 regions of the plane.  Let us collect here the notations which will be needed in the present chapter.
     A domain D is an open connected set of points.  That is, every point of D is the center of some δ-neighborhood, all of whose points are points of D; and any two points of D can be joined by a broken line having a finite number of segments, all of whose points are points of D.  A domain is bounded if all its points lie inside some square.
     A region R is a closed point set consisting of a bounded domain plus its boundary points.  We shall assume further that the boundary of R consists of a finite number of closed curves that do not cross themselves nor each other.  Note that the regions here defined and designated by the letter R are special cases of the more general ones of §3.1, Chapter 1.  In practical problems, R will usually be given in terms of its boundary curves.  For example, R might be the set of points between two concentric circumferences plus the points on the circumferences.  More frequently, we shall meet regions that can be most simply described by use of functions. Accordingly, we shall have a special notation for these.
     Let φ(x) and ψ(x) ∈ C in a ≤ x₁ ≤ b and φ(x) <ψ(x) in a < x < b.  Then the region R_x or R[a, b, φ(x), ψ(x)], is the region bounded by the curves   

x = a,     x = b,     y = φ(x),     y = ψ(x) 

If (x₁, y₁) is a point of R_x, then a ≤ x₁ ≤ b and φ(x₁)≤ y₁ ≤ ψ(x₁).  A line x = x₁, a  < x₁ < b cuts the boundary of R_x in just two points.  For example, the region R[-1, 1, − √(1 − x²), √(1 − x²)] is the circle x² + y² ≤ 1.  We could define in an obvious way a region R_y.  The region R described above as lying between two concentric circles is neither an R_x nor an R_y. It could be divided into four regions R_x, for example, by two vertical lines tangent to the inner circle.  These vertical lines would be counted twice, as the boundary of adjoining regions.
     A region R is simply connected if its boundary consists of a single closed curve.  The concept of the area of a region R will be assumed known.  Of course, the area of R_x is known from elementary calculus, and the area of R could be defined by use of a limiting process.  The diameter of a region R is the length of the longest line segment that joins two points of R.  In the case of a circle this coincides with the elementary notion of the diameter.  Observe that, if a region R varies so that its diameter approaches zero, then its area also approaches zero.  The converse is not true. 

1.2 DEFINITIONS
     We begin  by dividing a given region of R of area A into subregions.
As in the case of simple integrals, we introduce certain simplifying notations.
"
-Daivd Widder, Advanced Calculus

IN THE RED CORNER,  Euclid of Alexandria.

"The extremities of a surface are lines."
-Euclid, The Elements, Definition 6

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Notes

There are important differences between the two quotes.  We need the symbols necessary to operate algebraically, and to perform the Calculus.  To define arbitrary surfaces.  Our pencil is the function.  Still.  The surfaces we will be operating on in Chapter 6 of Widder's Advanced Calculus are precisely those which are defined in Euclid.  The definition is rigorous.  And so,

I have some questions for you, Modern Abstract Algebra.  And I don't mean Gauss.

1. What is the difference between a Dedekind section and Euclid's Definitions?
2. What is the relationship between infinite division and continuous measure?
   (Hint: No matter what anyone says, I can prove false by counterexample any mathematical argument which concludes by proof that I, Ryan, can measure.  All I need to do is disagree.  Likewise, a proof that I cannot measure.  All I need to do is say, "one apple."
   And likewise, I can prove any assertion that integral, unbroken measure does not exist, by waiting for a new instant in time, unbroken from the previous.  Or taking a step, some distance in some direction.  Left foot, right foot.  I defeat you, fake divisor-paradox!  Time for a nap.)
3. What is the logical status of a postulate?
4. I say, mathematics is not a logical circle.  Why?

I think, Mr. Modern, you do not answer these questions accurately.  Prove me wrong.  I'm asking for it.  See?

{Thus goad I a concept.
 Poke. 
 Thus}