Exploring Discontinuous Functions that are Continuous Except at Countable Points

When discussing the nature of functions on the real line, a fascinating example arises where a function is continuous at an uncountably infinite number of points yet discontinuous at a countable number of points. This article will delve into two such functions: a modified Dirichlet function and the function defined as Y x^n |x|.

1. The Modified Dirichlet Function

Let's consider the following modified Dirichlet function:

?f(x) 0, if x is irrational
? 1/q, if x p/q, where (gcd(p, q) 1) and (q 0)

Discontinuities: This function is discontinuous at all rational points along the real line. The reason for this is that in every arbitrarily small neighborhood of any rational number, there are infinitely many irrational numbers, and the function value at these irrational points is zero. Therefore, the function value at any rational point cannot be approximated by the values of the function in a neighborhood around it, leading to discontinuity.

Continuity at Irrational Points

However, the function is continuous at every irrational point x along the real line. To verify this, we note that the function is periodic with period 1. If x is irrational, then x - 1 is also irrational. Therefore, fx f(x - 1) 0. If x is rational, then x - 1 is also rational, and the function value is 1/q for some x p/q.
For every irrational point x within the interval (0,1), we need to show that the function is continuous. That is, for any arbitrarily small positive number epsilon, there exists a sufficiently small positive number delta such that for any rational number r within the interval (x - delta, x delta), the condition |f(x) - f(r)| epsilon holds.

Proof of Continuity

To demonstrate the continuity at every irrational point x, let (epsilon 0). Define (N 1/epsilon). The set of all rational numbers r p/q within (0,1) with q N is finite. Thus, there is a rational number among them that is the closest to x, denoted as q. Let (delta) be the distance between x and this closest rational number. Then, for any rational number r within ((x - delta, x delta)), we have q N, which implies |r - x| delta. Consequently, the function value at r is (frac{1}{q} epsilon). This completes the proof of the function's continuity at all irrational points in the interval (0,1), and thus throughout the real line.

Conclusion

Since the set of rational numbers is countably infinite, the function has exactly countably infinite points of discontinuity along the real line.

2. The Function (Y x^n |x|)

The second example is the function defined as Y x^n |x|. This function has a different behavior depending on the value of n.

Continuity: For n as a non-negative integer, the function is continuous everywhere on the real line. However, for negative values of n, the function is continuous at all irrational points and discontinuous at all rational points.

This function illustrates that by modifying the definition of a function, one can achieve the same goal of being continuous at an uncountable set of points while being discontinuous at a countable set.

In summary, both examples, the modified Dirichlet function and the function (Y x^n |x|), provide distinct insights into the nature of discontinuous functions continuous except at countable points, enriching our understanding of real analysis and function theory.