Proving Non-Riemann Integrability: A Function Defined Over Rational and Irrational Numbers

The topic of Riemann integrability often arises in the context of advanced mathematics, particularly when dealing with functions that are not continuous. In this article, we explore the non-integrability of a specific function that alternates between rational and irrational numbers over a defined interval. By understanding the measure and nature of discontinuities, we can determine the non-Riemann integrability of such a function.

Introduction

The function in question is defined as:

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

This function is particularly interesting because it essentially changes its value between two different linear functions based on the nature of the input (x) (rational or irrational). We will prove that this function is not Riemann integrable on the interval ([0, 1]). This proof will be based on the set of discontinuities and the measure of this set.

Step-by-Step Analysis

Step 1: Identify the Function

We start by clearly defining the function:

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

Step 2: Determine the Discontinuities

To check if (f(x)) is Riemann integrable, we must determine the set of its discontinuities. A function is Riemann integrable if the set of its discontinuities has measure zero. Here, we will show that (f(x)) is discontinuous at every point in the interval ([0, 1]).

Discontinuity at Rational Points

For any rational point (r) in ([0, 1]), the value of (f(r) r).

Consider a sequence of irrational numbers ({r_n}) converging to (r). For these points, (f(r_n) r_n - 1) approaches (r - 1) as (r_n) approaches (r). Thus, (f(r_n)) does not approach (f(r) r) as (r_n) approaches (r). This shows that (f(x)) is discontinuous at every rational point in ([0, 1]).

Discontinuity at Irrational Points

For any irrational point (s) in ([0, 1]), the value of (f(s) s - 1).

Consider a sequence of rational numbers ({q_n}) converging to (s). For these points, (f(q_n) q_n) approaches (s) as (q_n) approaches (s). Again, (f(q_n)) does not approach (f(s) s - 1) as (q_n) approaches (s). Thus, (f(x)) is also discontinuous at every irrational point in ([0, 1]).

Step 3: Measure of Discontinuities

Since (f(x)) is discontinuous at every point in the interval ([0, 1]), the set of discontinuities is the entire interval ([0, 1]). The measure of ([0, 1]) is 1, which is not zero.

Conclusion

Since the set of discontinuities of (f(x)) has positive measure, (f(x)) is not Riemann integrable on the interval ([0, 1]). Thus, we conclude that:

[ f(x) text{ is not Riemann integrable on } [0, 1] ]

General Proof for any Interval

The function (f: mathbb{R} to mathbb{R}) defined as:

[ f(x) begin{cases} x text{if } x in mathbb{Q} x - 1 text{if } x otin mathbb{Q} end{cases} ]

is not Riemann integrable on any interval ([a, b]), assuming without loss of generality that (a

By the Riemann-Lebesgue Theorem, if a function is bounded and has a set of discontinuities of measure zero, it is Riemann integrable. Here, the set of discontinuities of (f(x)) on ([a, b]) is ((b - a)), which is not zero. Therefore, (f(x)) is not Riemann integrable on ([a, b]).

Proof of Non-Continuity

To prove the function (f(x)) is nowhere continuous, fix a point (c in mathbb{R}) and suppose for contradiction that (f(x)) is continuous at (c). By the density of rational and irrational numbers, there exist sequences ({q_n}) of rational numbers and ({r_n}) of irrational numbers that both converge to (c).

Since we are assuming (f(x)) is continuous at (c), we must have that ({f(q_n)}) and ({f(r_n)}) both converge to (f(c)). However, ({f(q_n)} {q_n}) converges to (c), while ({f(r_n)} {r_n - 1}) converges to (c - 1). Therefore, (c eq c - 1), which is a contradiction. Hence, (f(x)) is nowhere continuous.

Final Remarks

The non-Riemann integrability of such a function highlights an interesting property of discontinuous functions and their behavior under the Riemann integral. Understanding these properties is crucial in advanced calculus and real analysis.