Problem 59 - Thomas Calculus

Question

Removable discontinuity Give an example of a function \(f(x)\) that is continuous for all values of \(x\) except \(x=2,\) where it has a removable discontinuity. Explain how you know that \(f\) is discontinuous at \(x=2,\) and how you know the discontinuity is removable. A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function $$ f(x)=\left\\{\begin{array}{ll}{1,} & {\text { if } x \text { is rational }} \\\ {0,} & {\text { if } x \text { is irrational }}\end{array}\right. $$ is discontinuous at every point. b. Is \(f\) right-continuous or left-continuous at any point?

Step-by-Step Solution

Answer

Function \(f(x)=\frac{x^2-4}{x-2}\) has a removable discontinuity at \(x=2\). Function with rational/irrational pair is discontinuous everywhere and not left/right continuous.

Step 1: Define a function with a removable discontinuity

Consider the function \[ f(x) = \frac{x^2 - 4}{x - 2} \]This function appears to be undefined at \(x=2\) because the denominator \(x-2\) becomes zero.

Step 2: Simplify the function to remove the discontinuity

Factor the numerator:\[ x^2 - 4 = (x-2)(x+2) \]Thus, \[ f(x) = \frac{(x-2)(x+2)}{x-2} = x+2, \quad x eq 2 \]The single factor of \(x-2\) cancels out, leaving \(x+2\), which is defined for all \(x\).

Step 3: Explain the removable discontinuity

Initially, \(f(x)\) is not defined at \(x=2\), indicating a discontinuity. However, after simplification, \(f(x)\) can be redefined to make it continuous by defining \(f(2) = 4\). This local adjustment is called a removable discontinuity.

Step 4: Analyze the function for pointwise discontinuity

Consider the function,\[ f(x) = \begin{cases} 1, & \text{if } x \text{ is rational} \ 0, & \text{if } x \text{ is irrational} \end{cases} \]At any real number, both rational and irrational sequences can converge to that point, leading to two different limits. Thus, \(f(x)\) has no point of continuity and is discontinuous everywhere.

Step 5: Evaluate continuity type (right or left)

For any real number \(x_0\), whether from the right or left, the limits will result in a rational and irrational approach, causing no single limit value to coincide with the function, thus \(f(x)\) is neither right-continuous nor left-continuous anywhere.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\) is not zero. Some common examples of rational numbers include:

• Whole numbers like 3 and 7
• Fractions like \(\frac{1}{2}\) or \(-\frac{3}{4}\)
• Negative numbers such as -5

Rational numbers are plentiful on the number line, densely packing every interval. In terms of the example given in the exercise, when \(x\) is rational, \(f(x) = 1\), representing a constant value at those points.

Irrational Numbers

Irrational numbers, on the other hand, cannot be expressed as simple fractions. Examples include \(\sqrt{2}\), \(\pi\), and \(e\). These numbers have non-repeating, non-terminating decimal expansions.
They also fill the number line densely, present between any two rationals, and form a basis for continuous, yet complex, behaviors in functions.

• Unlike rational numbers, they create more complex sequences and limits, as they can be approached but never exactly reached by simple fractions.
• In the exercise example, when \(x\) is irrational, \(f(x) = 0\), showing different behavior compared to rational inputs.

Discontinuous Functions

Discontinuous functions are those which have one or more breaks, jumps, or undefined points within their domain. A classic type of discontinuity is the removable kind, where a function can be redefined at a certain point to make it continuous.
The first part of the exercise demonstrates this with \(f(x) = \frac{x^2 - 4}{x - 2}\), where the discontinuity at \(x = 2\) is removable by simplifying and redefining the function.
For \(f(x)\) used throughout the exercise,

• The function switches between distinct values for rational and irrational numbers, exemplifying a discontinuous behavior across all points.
• This leaves no continuity between any two points due to its dependency on number type.

Right-Continuity

Right-continuity refers to a function being continuous from the right side at a specific point \(x_0\). It is defined such that as \(x\) approaches \(x_0\) from the right, the function approaches the value \(f(x_0)\).
For a function to be right-continuous, the limit from the positive side must exist and equal the function value at the point.

• In the provided exercise, due to the mixed nature (rational vs. irrational), there is no such value that both sides will agree upon; hence, the function is not right-continuous anywhere.

Left-Continuity

Left-continuity mirrors that of right-continuity, requiring the function to maintain continuity as \(x\) approaches from the left side. A function is left-continuous at \(x_0\) if the limit from the negative side coincides with \(f(x_0)\).
In the example from the exercise, like right-continuity, \(f(x)\) fails to meet the left-continuity criteria at any point due to differing results for rational and irrational inputs approaching any given point.

• Thus, no consistent approach from the left or right side satisfies the continuity requirements.