Points of Discontinuity for the Dirichlet Function

The Dirichlet function, denoted as (f(x)), is defined as:

[f(x) begin{cases} 1 text{if } x text{ is rational} 0 text{if } x text{ is irrational} end{cases}]

Letrsquo;s analyze the points of discontinuity for this function. To determine where the function is discontinuous, we need to understand the concept of continuity and how it applies to the Dirichlet function.

Continuity Analysis

A function (f) is considered continuous at a point (c) if the limit of the function as (x) approaches (c) equals the value of the function at (c). Mathematically, this means:

[lim_{x to c} f(x) f(c)]

However, for the Dirichlet function, we need to evaluate (f(c)) and the limit behavior as (x) approaches (c).

Definition of (f(c))

If (c) is rational, then (f(c) 1). Otherwise, if (c) is irrational, then (f(c) 0).

Limit Evaluation

To evaluate the limit (lim_{x to c} f(x)), we consider the nature of the real numbers and the density of rational and irrational numbers around any point emc/em.

Since the set of rational numbers and the set of irrational numbers are both dense in the real numbers, as (x) approaches (c), (f(x)) can take on both values 0 and 1. This means:

For rational (x) approaching (c), (f(x) 1). For irrational (x) approaching (c), (f(x) 0).

The limit does not settle to a single value because the function emoscillates between 0 and 1/em as (x) approaches (c). In other words:

[lim_{x to c} f(x) eq f(c)]

Conclusion

Since the limit does not equal (f(c)) at any point (c) on the real number line, the Dirichlet function is discontinuous at every point in (mathbb{R}).

So, we can conclude:

The function (f) is discontinuous at every point (c) in (mathbb{R}).

Thus, every real number is a point of discontinuity of the Dirichlet function.

Further Reading: The dense property of real numbers states that between any two real numbers, there exists both a rational and an irrational number. This property is crucial in understanding why the Dirichlet function oscillates because of the interweaving nature of rational and irrational numbers.