Continuous Transformations in Analysis : With an Introduction to Algebraic Topology The general objective of this treatise is to give a systematic presenta­ tion of some of the topological and measure theoretical foundations of the theory of real valued functions of several real variables with particular emphasis upon a line of thought initiated by BANACH GEOCZE LEBESGUE TONELLI and VITALI. To indicate a basic feature in this line of thought let us consider a real valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago BANACH and VITALI observed that the fundamental concepts of bounded variation absolute continuity and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n space Rff where T is given by a system of equations of the form 1 (1 ff) X I U . . . tt . and n is an arbitrary positive integer. Accordingly these geometrical descriptions can be used to define for continuous transformations in Euclidean n space Rff n dimensional concepts 01 bounded variation and absolute continuity and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed generalized and modified by many mathematicians and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE LEBESGUE and TONELLI. I. Background in Topology § I.1. Survey of general topology § I.2. Survey of Euclidean spaces § I.3. Survey of Abelian groups § I.4. Mayer complexes § I.5. Formal complexes § I.6. General cohomology theory § I.7. Cohomology groups in Euclidean spaces II. Topological study of continuous transformations in Rn § II.1. Orientation in Rn § II.2. The topological index § II.3. Multiplicity functions and index functions III. Background in Analysis § III.1. Survey of functions of real variables § III.2. Functions of open intervals in Rn IV. Bounded variation and absolute continuity in Rn § IV.1. Measurability questions § IV.2. Bounded variation and absolute continuity with respect to a base function § IV.3. Bounded variation and absolute continuity with respect to a multiplicity function § IV.4. Essential bounded variation and absolute continuity § IV.5. Bounded variation and absolute continuity in the Banach sense V. Differentiable transformations in Rn § V.1. Continuous transformations in R1 § V.2. Local approximations in Rn § V.3. Special classes of differentiable transformations in Rn VI. Continuous transformations in R2 § VI.1. The topological index in R2 § VI.2. Special features of continuous transformations in R2 § VI.3. Special classes of differentiable transformations in R2.