Question

Construct an increasing function on \(\mathbb{R}\) whose set of discontinuities is \(\mathbb{Q}\).

Short answer

The function \( f(x) = \sum_{q \in \mathbb{Q}, q \leq x} 2^{-n(q)} \) achieves this.

Step-by-step solution

Understand the requirements

We need to construct a function that is defined over the set of real numbers \( \mathbb{R} \) and is increasing. Additionally, the function must be discontinuous at every rational number and continuous at every irrational number.

Consider a candidate function

A useful approach is to leverage the properties of dense sets. Rational numbers \( \mathbb{Q} \) are dense in \( \mathbb{R} \), and they are also countable. Consider the function \( f(x) = \sum_{q \in \mathbb{Q}, q \leq x} 2^{-n(q)} \), where \( n(q) \) is the position of \( q \) in some enumeration of the rationals.

Analyze the behavior at rational points

At a point \( x = q_0 \in \mathbb{Q} \), there is a jump discontinuity. Before this point, the sum considers all smaller rationals. At the exact point \( q_0 \), a new term \( 2^{-n(q_0)} \) is added, causing the function to jump.

Analyze the behavior at irrational points

At a point \( x \) being irrational, the function sum essentially captures only finite rationals, and as \( x \) comes closer to any particular rational, there’s no single rational for which a jump is added because no finite accumulation can occur.

Verify the increasing nature of the function

For any \( x_1 < x_2 \), \( f(x_2) \) includes additional terms from \( f(x_1) \) thus making \( f(x_2) \geq f(x_1) \), thereby ensuring the function is indeed increasing.

Conclude the properties of the function

The function \( f(x) \) is increasing, discontinuous at every rational point by construction, and continuous at every irrational point, satisfying the problem's requirements.

Key concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. These numbers are a vital part of mathematics and are denoted by the symbol \( \mathbb{Q} \). They take the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
\( \mathbb{Q} \) is dense in \( \mathbb{R} \), meaning between any two real numbers, there is a rational number. This property plays a crucial role when dealing with continuity and discontinuity within functions. When constructing a function with discontinuities at precisely all rational numbers, such as in the given exercise, this density creates predictable discontinuous behavior wherever these rational numbers occur. Also, rational numbers are countable, meaning they can be listed in a sequence like \( q_1, q_2, q_3, \ldots \).
As a result, enumerating them becomes straightforward, enabling the incorporation of discrete elements whenever tackling problems around discontinuity and increasing functions.

Dense Sets

The concept of dense sets is central in understanding many properties of functions. A dense set in real numbers means that between any two real numbers, no matter how close, there is another point from the dense set.
One primary example of a dense set is the set of rational numbers \( \mathbb{Q} \). This property is crucial for constructing functions with specified behaviors, such as discontinuities at irrational numbers. For the function given in the exercise, using the denseness of rationals allows for the construction of a function that is specifically discontinuous at every rational number.

• Because rationals are dense, any interval on the real number line (no matter how small) contains infinitely many rational numbers.
• This ensures that any function intended to be discontinuous everywhere there is a rational number behaves predictably across \( \mathbb{R} \).
• It allows the function to leverage the predictable nature of rational interspersing to precisely introduce discontinuities where required.

Understanding dense sets is fundamental for mathematical analysis, especially when working to match specific functional behaviors to the types of numbers (rational or irrational) being considered.

Increasing Functions

An increasing function is one in which, if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). Essentially, as you move along the function from left to right, the value of the function does not decrease.
This characteristic is vital in mathematical analysis and was a requirement for the function constructed in the original exercise. The function had to remain increasing even as it exhibited discontinuities at each rational number. Here's how it was achieved:

• For any two real numbers \( x_1 < x_2 \), the sum that defines the function, \( f(x) = \sum_{q \in \mathbb{Q}, q \leq x} 2^{-n(q)} \), includes every rational below \( x_2 \) that is also below \( x_1 \).
• As \( x \) progresses from \( x_1 \) to \( x_2 \), new rational terms are progressively added.
• The function will always add terms, or at least maintain current terms, consistently ensuring that \( f(x_2) \geq f(x_1) \).

Therefore, despite the function's increasing nature, each time it meets a rational number, it meets the criteria for a discontinuity because of the jumps introduced by the added terms \( 2^{-n(q)} \). This maintains the ordered nature of \( f \) while precisely controlling where discontinuities occur.