S.SANTHANAM

  Riemann integration [ edit ] Main article:  Riemann integral The Riemann integral is defined in terms of  Riemann sums  of functions with respect to tagged partitions of an interval. Let  [ � , � ]  be a  closed interval  of the real line; then a  tagged partition   �  of  [ � , � ]  is a finite sequence � = � 0 ≤ � 1 ≤ � 1 ≤ � 2 ≤ � 2 ≤ ⋯ ≤ � � − 1 ≤ � � ≤ � � = � . This partitions the interval  [ � , � ]  into  �  sub-intervals  [ � � − 1 , � � ]  indexed by  � = 1 , … , � , each of which is "tagged" with a distinguished point  � � ∈ [ � � − 1 , � � ] . For a function  �  bounded on  [ � , � ] , we define the  Riemann sum  of  �  with respect to tagged partition  �  as ∑ � = 1 � � ( � � ) Δ � , where  Δ � = � � − � � − 1  is the width of sub-interval  � . Thus, each term of the sum is the area of a rectangle with height ...

  1.3 THE REAL LINE  One of our objectives is to develop rigorously the concepts of limit, continuity, differentiability, and integrability, which you have seen in calculus. To do this requires a better understanding of the real numbers than is provided in calculus. The purpose of this section is to develop this understanding. Since the utility of the concepts introduced here will not become apparent until we are well into the study of limits and continuity, you should reserve judgment on their value until they are applied. As this occurs, you should reread the applicable parts of this section. This applies especially to the concept of an open covering and to the Heine–Borel and Bolzano–Weierstrass theorems, which will seem mysterious at first. We assume that you are familiar with the geometric interpretation of the real numbers as points on a line. We will not prove that this interpretation is legitimate, for two reasons: (1) the proof requires an excursion into the foundatio...

  Quasi-isometries [ edit ] Main article:  Quasi-isometry A  quasi-isometry  is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example,  � 2  and its subspace  � 2  are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in  geometric group theory : the  Švarc–Milnor lemma  states that all spaces on which a group  acts geometrically  are quasi-isometric. [15] Formally, the map  � : � 1 → � 2  is a  quasi-isometric embedding  if there exist constants  A  ≥ 1  and  B  ≥ 0  such that 1 � � 2 ( � ( � ) , � ( � ) ) − � ≤ � 1 ( � , � ) ≤ � � 2 ( � ( � ) , � ( � ) ) + �  for all  � , � ∈ � 1 . It is a  quasi-isometry  if in addition it is  quasi-surjective , i.e. there is a constant  C  ≥ 0  such that...