Problem 75 - Calculus Early Transcendentals: Pearson New International Edition

Question

75\. Show that if \(g(x)=|f(x)|\) is continuous it is not necessarily true that \(f(x)\) is continuous.

Step-by-Step Solution

Answer

If \(g(x) = |f(x)|\) is continuous, \(f(x)\) may still be discontinuous; see \(f(x) = -1\) for \(x=0\), \(1\) for \(x\neq 0\).

Step 1: Understanding the Problem

To solve this problem, we need to provide an example where the function \(g(x) = |f(x)|\) is continuous, but the function \(f(x)\) itself is not continuous.

Step 2: Choose a Function with Discontinuity

Let's consider the function \(f(x)\) defined as follows: \[ f(x) = \begin{cases} -1, & \text{if } x = 0 \ 1, & \text{if } x eq 0 \end{cases} \] This function is discontinuous at \(x = 0\) because the limit as \(x\) approaches 0 does not equal the value of \(f(0)\).

Step 3: Construct \(g(x)\) using Absolute Value

We define \(g(x) = |f(x)|\). For our chosen \(f(x)\), we have: \[ g(x) = |f(x)| = \begin{cases} 1, & \text{if } x = 0 \ 1, & \text{if } x eq 0 \end{cases} \] Simplifying gives \(g(x) = 1\) for all \(x\), which is a constant function.

Step 4: Prove Continuity of \(g(x)\)

The function \(g(x)\) is constant, and constant functions are continuous everywhere. Therefore, \(g(x)\) is continuous for all \(x\).

Step 5: Conclude the Example

We have shown that \(g(x) = |f(x)|\) is continuous but \(f(x)\) is discontinuous at \(x = 0\). This example demonstrates that \(g(x) = |f(x)|\) being continuous does not imply that \(f(x)\) is continuous.

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that helps understand how far a number is from zero on the number line, regardless of direction. When we talk about the absolute value of a function, we are essentially transforming any negative outputs to positive ones, while leaving positive outputs unchanged.
For a function like \(f(x)\), its absolute value is represented as \(|f(x)|\). This transformation ensures that the output of \(|f(x)|\) is never negative, since the absolute value strips away any negative sign.

• If \(f(x)\) is positive or zero, then \(|f(x)| = f(x)\).
• If \(f(x)\) is negative, then \(|f(x)| = -f(x)\), which results in a positive value.

This operation can sometimes help create a continuous function from a discontinuous one, depending on how the negativity is distributed across the domain, as shown in the example provided where \(g(x) = |f(x)| \) becomes continuous while \(f(x)\) is discontinuous.

Discontinuous Function

A function is said to be discontinuous at a point if there is a break, jump, or hole in its graph at that particular point. Specifically, for a function \(f(x)\), it is discontinuous at a point if the following conditions are not met:

• The limit of \(f(x)\) as \(x\) approaches a point exists.
• The value of the function at that point exists.
• The limit value at that point equals the function value.

In our example, \(f(x)\) is defined differently based on the value of \(x\). At \(x = 0\), \(f(x)\) experiences a jump discontinuity, as the value of the function changes abruptly between \(-1\) and \(1\).
This discrepancy at \(x = 0\) prevents \(f(x)\) from having a continuous path without breaks on the graph.

Constant Function

A constant function is one of the simplest types of functions, where the output value remains the same regardless of the input. Mathematically, it is expressed as \(g(x) = c\), where \(c\) is a constant.
In our example, after applying the absolute value, the function \(g(x) = |f(x)|\) simplifies to \(g(x) = 1\). Here, for every value of \(x\), the result is consistently \(1\), showcasing the property of a constant function.

• Constant functions represent horizontal lines on a graph.
• They have no fluctuations, making them smooth and uniform across their domain.

Moreover, constant functions are inherently continuous everywhere because there are no jumps, breaks, or irregularities as you move along the x-axis.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as it approaches a certain point. Understanding limits is crucial to discussing the continuity of a function.
For a limit to exist, the values of \(f(x)\) as \(x\) approaches a particular point from both the left and right must converge to the same value. This value is what we call the limit of the function as \(x\) approaches that point.
In the provided exercise, we saw that \(f(x)\) fails to meet this criterion at \(x = 0\), making it discontinuous. However, the absolute value transformation \(g(x) = |f(x)|\) effectively sidesteps this discontinuity by ensuring the left and right approach result in the same constant value.

• Limits help determine the continuity of a function at a specific point.
• They are key in defining derivatives and integrals in calculus.