Understanding the Weierstrass Function: A Guide to Nowhere Continuous Functions

Weierstrass function is a mathematical function that is used to define the concept of a continuous function. It was introduced by Karl Weierstrass, a German mathematician, in the late 19th century. The function is defined as follows:

f(x) = 0 if x is rational (x = a/b, where a and b are integers and b is non-zero)

f(x) = 1/x if x is irrational (x cannot be expressed as a ratio of integers)

The Weierstrass function is an example of a nowhere continuous function, meaning that it has no points at which it is continuous. This is because the function is constantly changing its value as x changes, and there are no points at which the function is defined at all times.

The Weierstrass function has several important properties, including:

1. It is continuous at every point in the rational numbers, but not at any point in the irrational numbers.
2. It is differentiable at every point in the rational numbers, but not at any point in the irrational numbers.
3. It has a finite limit at every point in the rational numbers, but does not have a finite limit at any point in the irrational numbers.
4. It is bounded above and below by the functions 0 and 1/x, respectively.

The Weierstrass function has many applications in mathematics, particularly in the field of real analysis. It is used to demonstrate the existence of nowhere continuous functions, which are functions that have no points at which they are continuous. It also has applications in other areas of mathematics, such as topology and measure theory.