Problem 60 - Calculus Early Transcendentals

Question

For what values of \(x\) is \(g\) continuous? $$g(\mathrm{x})=\left\\{\begin{array}{ll}{0} & {\text { if } x \text { is rational }} \\ {x} & {\text { if } x \text { is irrational }}\end{array}\right.$$

Step-by-Step Solution

Answer

The function \(g(x)\) is not continuous for any real number \(x\).

Step 1: Understanding the piecewise function

The function \(g(x)\) is defined as a piecewise function, with two parts: \(g(x) = 0\) if \(x\) is rational, and \(g(x) = x\) if \(x\) is irrational. Our task is to determine for which values of \(x\) this function is continuous.

Step 2: Defining continuity of a function

A function \(g(x)\) is continuous at a point \(a\) if the limit of \(g(x)\) as \(x\) approaches \(a\) from all directions equals \(g(a)\), i.e., \( \lim_{x \to a} g(x) = g(a) \).

Step 3: Analyzing continuity at rational points

If \(x=a\) is rational, then \(g(a) = 0\). For continuity, both limits \( \lim_{x \to a^+} g(x) \) and \( \lim_{x \to a^-} g(x) \) must equal \(0\). However, \(g(x)\) approaches \(a\) for irrational \(x\) near \(a\), and approaches \(0\) for rational \(x\), indicating \(g(x)\) is not continuous at rational points.

Step 4: Analyzing continuity at irrational points

If \(x=a\) is irrational, then \(g(a) = a\). For continuity, \( \lim_{x \to a} g(x) \) must equal \(a\). For any \(x\) arbitrarily close to \(a\), \(g(x)\) alternates between \(0\) (for rational \(x\)) and \(x\) (irrational \(x\)), which prevents the limit from consistently equaling \(a\). Consequently, \(g(x)\) is not continuous at irrational \(x\) either.

Step 5: Conclusion on function continuity

Since \(g(x)\) is discontinuous at both rational and irrational points, it is not continuous for any real number \(x\).

Key Concepts

Piecewise Functions

A piecewise function is a type of function defined by multiple sub-functions, each operating over a particular part of the domain. In the case of our function \( g(x) \), it is defined with two distinct rules based on whether \( x \) is rational or irrational. This means:

• If \( x \) is a rational number, the output is always \( 0 \).
• If \( x \) is an irrational number, the output is \( x \) itself.

This dual definition is why we call \( g(x) \) a piecewise function—it pieces together different rules based on the characteristics of \( x \).
Why do we use piecewise functions? They are particularly helpful in modeling real-world situations where there are different conditions or rules that apply in different scenarios. Think of it like a restaurant menu where different items are available at breakfast and dinner times.
In mathematics, piecewise functions can sometimes create challenges such as discontinuities or unexpected behavior at certain points.

Discontinuity

Discontinuity in a function means there are points where the function does not behave smoothly, or the limit does not exist as expected. In simpler terms, there is a "break" in the graph of the function.
For the function \( g(x) \), determining points of continuity requires examining how the function behaves around any \( x \). We check if \( \lim_{x \to a} g(x) = g(a) \) for every \( a \) in the domain.
Let's explore:

• Rational \( x \): The value jumps since \( g(x) = 0 \) at rational numbers, but changes to \( x \) at irrational numbers adjacent to \( a \). This back-and-forth between 0 and \( a \) at small intervals creates a break.
• Irrational \( x \): Here, \( g(a) \) equals \( a \), but any close-by \( x \), which may be rational, makes \( g(x) = 0 \). Hence, the limit near these points differs from the actual function value.

Thus, \( g(x) \) is never continuous because its behavior near any \( a \) does not match the value at exactly \( a \). Each point, whether rational or irrational, causes discontinuity.

Rational and Irrational Numbers

Understanding rational and irrational numbers is essential when dealing with piecewise functions like \( g(x) \). These numbers dictate which rule of the piecewise function to apply.

• Rational Numbers: These are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). Examples include \( 1/2, 0.75, \) or any finite or repeating decimal.
• Irrational Numbers: These cannot be expressed as a simple fraction. Their decimal form is non-repeating and infinite. Famous examples are \( \pi \) and \( \sqrt{2} \).

The function \( g(x) \) reveals how these two different sets of numbers behave uniquely in mathematical problems. Using different definitions for rational and irrational values introduces complexity, particularly in determining continuity. Each type of number leads to transitions in the function’s output that exemplify the intricacies of mathematical behavior, such as discontinuity when different properties of numbers intersect.​