91Ó°ÊÓ

Complex variables are fundamental in studying functions like the given function, which takes complex numbers as input and gives complex numbers as output. In the context of uniform continuity, complex variables can exhibit behaviors that greatly differ from real variables. In this exercise, the focus is on the complex variable \( z \) within the open unit disc \( |z|<1 \), which is a subset of the complex plane.

A complex variable is expressed in the form \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). Manipulating these variables often involves considering both their real and imaginary parts. Additionally, the concept of a limit and continuity for complex variables requires understanding paths and regions in the complex plane, which plays a key role when discussing uniform continuity.

Function sequences are an instrumental tool in exploring the concept of continuity, especially uniform continuity. A sequence in mathematics is an ordered list of numbers, and similarly, a function sequence is an ordered list of function values. In this exercise, we have sequences \( (x_n) \) and \( (y_n) \), where both sequences converge to a common limit as \( n \) approaches infinity. By applying the function \( f(z) = 1 / (1 - z) \) to these sequences, we create a new sequence of numbers representing the function values at each term of the original sequences.

The function values for the sequences, \( f(x_n) \) and \( f(y_n) \), help us illustrate and understand the behavior of the function. If the difference between the function values from two converging sequences does not shrink to zero, the function fails to be uniformly continuous. This is a crucial aspect when analyzing functions over complex variables and is used to demonstrate the lack of uniform continuity in our exercise.

Limit points, or accumulation points, are indispensable when discussing uniform continuity. A limit point of a set is a point that can be approached by the elements of the set as closely as desired, which does not necessarily mean that the point is within the set itself. For functions defined on the complex plane, like \( f(z) = 1 / (1 - z) \), studying their behavior near the limit points is essential to understanding continuity.

In the case of this exercise, both sequences \( (x_n) \) and \( (y_n) \) converge to the same limit point, zero. Nonetheless, due to the nature of the function, the difference in the function values are not negligible even as the points grow arbitrarily close to each other, indicating the function is not uniformly continuous around that limit point. Thus, the properties of limit points, along with function sequences, serve as a strategic approach to dissecting the uniform continuity of complex functions.