# Continuous transformations in R2

• P. V. Reichelderfer
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (GL, volume 75)

## Abstract

Throughout Part VI, we shall be concerned with bounded continuous transformations
$$T:D \to {R^2}$$
(1)
where D is a bounded domain in R2. It will be convenient to use certain alternative representations for T. Note that the points of R 2 are ordered pairs (x, y) of real numbers x, y. Hence we can associate with the point (x, y) of R2 the complex number z = x + iy. In the sequel, the terms “complex number” and “point of R2” will be used interchangeably. Using complex numbers, the transformation (1) can be represented in the form
$$T:z = T\left( w \right),w \in D$$
(2)
where z is the image point of w under T. Thus T is thought of now as a bounded, continuous, real or complex-valued function of the complex number wD. If we set w = u + iv, z = x + iy, where u, v, x, y are real numbers, then we obtain for T the representation
$$T:x = x\left( {u,v} \right),y = y\left( {u,v} \right),\left( {u,v} \right) \in D$$
(3)
where the coordinate functions x(u, v), y(u, v) are bounded, continuous, real-valued functions in D.

### Keywords

Clarification Proal Prool