Question

Let $$ f(x)=\left\\{\begin{aligned} x & \text { if } x \text { is rational } \\ -x & \text { if } x \text { is irrational } \end{aligned}\right. $$ Sketch the graph of this function as best you can and decide where it is continuous.

Short answer

The function is not continuous anywhere on the real line.

Step-by-step solution

Understanding the function

The given function is defined piecewise: it outputs \( x \) if \( x \) is a rational number and \( -x \) if \( x \) is an irrational number. The function behaves differently depending on the nature of \( x \).

Graphing the function - Rational Black Dots

For rational numbers, plot the graph of \( f(x) = x \). This graph would look like a straight line passing through the origin with a slope of 1. However, plot these points as black dots since they are not connected continuously.

Graphing the function - Irrational Hollow Dots

For irrational numbers, plot the graph of \( f(x) = -x \). This graph would look like a straight line as well, passing through the origin but with a slope of -1. Also, plot these points as hollow dots since they won't create a continuous line.

Analyzing Continuity

A function is continuous at a point if the limit of \( f(x) \) as \( x \) approaches any point \( c \) from both sides equals \( f(c) \). This function does not have this property at any point because rational and irrational numbers are densely mixed on the number line and give different values near all \( c \).

Conclusion on Continuity

Since the function changes its output between \( x \) and \( -x \) for any arbitrarily small neighborhood around any number, the function is not continuous everywhere on the real line.

Key concepts

Piecewise Functions

Piecewise functions are fascinating because they allow a function to have different rules or formulas depending on the input value. This makes them especially useful in math to describe situations where a single expression is insufficient to capture the behavior of a function across its entire domain. In the exercise, the function is defined as piecewise, meaning the output depends on whether the input is a rational or an irrational number.

- For rational inputs, the function outputs the input itself. - For irrational inputs, the function outputs the negative of the input.
The concept of piecewise functions is that each piece covers a specific part of the input domain, here distinguished by the nature of rationality and irrationality. This specificity allows different behaviors on different intervals or categories of numbers, under the rule governing that piece. Understanding piecewise functions helps in visualizing and separating different parts of a function based on distinctive characteristics of the input.

Rational and Irrational Numbers

Rational and irrational numbers are two fundamental categories of numbers in mathematics. Understanding these concepts is crucial in dealing with piecewise functions, like the one in our exercise. Rational numbers are numbers that can be expressed as the quotient of two integers. Simply put, they can be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers, with \(b eq 0\). Examples include 1/2, 3, and -4.

On the other hand, irrational numbers cannot be expressed simply as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Classic examples include \(\sqrt{2}\), \pi\, and \(e\).

In the context of the given function, these two types of numbers determine how the function behaves: the output matches the input if it's rational, or flips sign if it's irrational, showing clearly how a function can leverage the properties of rationality to define distinct operations.

Graphing Functions

Graphing functions is an analytical method to visualize the behavior of a function over a range of values. In the exercise, graphing helps us to understand the discontinuous nature of the function. When graphing a function involving rational and irrational inputs, as in this case, different methods of plotting come into play.

- For rational numbers, plot points where the function behaves like \(f(x) = x\). Imagine each rational point as a black dot on the line \(y = x\).- For irrational numbers, the graph plots points where the function behaves like \(f(x) = -x\). Here, visualize each irrational point as a hollow dot on the line \(y = -x\).

The critical insight drawn from graphing this piecewise function is the lack of continuity. No line connects the two sets of points, because rational and irrational numbers alternate on the number line, causing the output to jump between two values, making it impossible for any segment between them to connect. Graphing reveals the true nature of this discontinuity.