Problem 75 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\( f(x) \) can be discontinuous even if \( g(x) = |f(x)| \) is continuous.

Step 1: Understand the Problem

We need to show that the function \( f(x) \) being continuous isn't a necessary condition for \( g(x) = |f(x)| \) to be continuous. This means even if \( g(x) \) is continuous, \( f(x) \) might not be.

Step 2: Define a Counterexample

Consider the function \( f(x) = \begin{cases} -1, & x < 0 \ 1, & x \geq 0 \end{cases} \). The function \( f(x) \) is not continuous at \( x = 0 \), because the limit from the left \( f(x) \to -1 \) and from the right \( f(x) \to 1 \) are not equal.

Step 3: Examine the Absolute Function

Calculate \( g(x) = |f(x)| \). For \( x < 0, \ g(x) = |-1| = 1 \) and for \( x \geq 0, \ g(x) = |1| = 1 \). Hence, \( g(x) = 1 \) for all \( x \).

Step 4: Check Continuity of g(x)

The function \( g(x) = 1 \) is continuous everywhere, since it is a constant function. Constants are always continuous.

Step 5: Conclusion from Counterexample

Since \( g(x) = |f(x)| \) is continuous even though \( f(x) \) is not, this demonstrates that \( f(x) \) does not need to be continuous for \( g(x) \) to maintain continuity.

Key Concepts

Absolute Value Function

The absolute value function takes any real number and returns its non-negative value. Mathematically, it's defined as \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). This function essentially measures the distance of a number from zero on the number line, disregarding direction.
When dealing with more complex functions like \(g(x) = |f(x)|\), it means we take the function \(f(x)\) and apply this absolute value rule to each output. If \(f(x)\) is a continuous function, its absolute values can form another function. However, the continuity of \(|f(x)|\) doesn't automatically imply that \(f(x)\) itself is continuous, as demonstrated in the problem.
Using \(g(x) = |f(x)|\), even when \(f(x)\) changes sign, \(g(x)\) remains smooth, as it converts all negative outputs to positive, thus potentially obscuring discontinuities in \(f(x)\).

Counterexample

In mathematics, a counterexample is a specific case that disproves a statement or proposition. For the statement, "If \(g(x)=|f(x)|\) is continuous, then \(f(x)\) must be continuous," we need a function that shows this isn't true.
Consider the function \(f(x) = \begin{cases} -1, & x < 0 \ 1, & x \geq 0 \end{cases}\). This function is not continuous at \(x=0\) because the left-hand limit and right-hand limit are different. Yet, when we take its absolute value, we get \(g(x) = |f(x)| = 1\), which is a constant function and continuous everywhere.
Thus, this counterexample effectively demonstrates that the continuity of \(g(x)\) does not guarantee the continuity of \(f(x)\). It's why counterexamples are crucial in mathematical reasoning; they highlight exceptions and clarify boundaries in logical statements.

Constant Function

A constant function is a simple and predictable function where the output value is the same for any input value. Formally, if \(g(x) = c\) for all \(x\), where \(c\) is a constant, then \(g(x)\) is a constant function.
Constant functions are inherently continuous. This is because, by definition, there are no jumps, breaks, or fluctuations in their value. The graph of a constant function is a perfectly straight, horizontal line, which means it passes the test for continuity at every point, as it has no gaps.
In the exercise, the absolute value of \(f(x)\) results in a constant function \(g(x) = 1\) everywhere. Despite \(f(x)\) not being continuous at \(x=0\), the resulting constant \(g(x)\) remains continuous. This showcases how absolute value, by transforming the outputs, can mask discontinuities in the original function.

Limits and Continuity

Limits and continuity are fundamental concepts in calculus that help us understand the behavior of functions at specific points or over intervals. A function \(f(x)\) is continuous at a point \(x = a\) if:

• The function is defined at \(x=a\).
• The limit of \(f(x)\) as \(x\) approaches \(a\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(a\) equals \(f(a)\).

If any of these conditions fails, the function has a discontinuity at that point. In the original problem, \(f(x)\) fails at continuity because its left and right limits at \(x=0\) don't match. However, applying the absolute value leads to \(g(x) = |f(x)|\), which is continuous everywhere as its value is constant.
This illustrates how applying transformations like absolute value can influence continuity, smoothing out discontinuities in the generated function despite visible ones in the original.