Question

Let $$g(x)=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\\-1 & \text { if } x<0\end{array}\right.$$ a. Write a formula for \(|g(x)|\) b. Is \(g\) continuous at \(x=0 ?\) Explain. c. Is \(|g|\) continuous at \(x=0 ?\) Explain. d. For any function \(f,\) if \(|f|\) is continuous at \(a,\) does it necessarily follow that \(f\) is continuous at \(a ?\) Explain.

Short answer

The formula for the absolute value of g(x) is |g(x)| = 1, always. b) Is g(x) continuous at x=0? No, g(x) is not continuous at x=0. c) Is the absolute value of g(x) continuous at x=0? Yes, the absolute value of g(x) is continuous at x=0. d) Is it true that whenever the absolute value of a function f is continuous at a, the function itself must be continuous at a? No, this statement is false. The given counterexample (g(x)) demonstrates that a function can have a continuous absolute value at a point but not be continuous itself at that point.

Step-by-step solution

Determine the formula for |g(x)|

To find the absolute value of g(x), let's analyze both of its conditions. - When x is greater than or equal to 0, g(x) = 1. The absolute value of 1 is still 1, so |g(x)| = 1 for x >= 0. - Similarly, when x is less than 0, g(x) = -1. The absolute value of -1 is 1, so |g(x)| = 1 for x < 0. Having analyzed both conditions, we can write |g(x)| as a piecewise function: $$|g(x)|=\left\{\begin{array}{ll}1 & \text { if } x \geq 0 \\1 & \text { if } x<0\end{array}\right.$$ That is, |g(x)| is always 1, no matter the value of x.

Determine if g(x) is continuous at x=0

A function is continuous at a point if the following conditions hold: - The function is defined at that point. - The left and right limits of the function at that point exist and are equal. - The function f(x) approaches the same limit as we approach the point from both the left and the right, i.e., f(x) = lim(x→a) f(x). For g(x), let's analyze its behavior around x=0. As x approaches 0 from the right, g(x) = 1. When x approaches 0 from the left, g(x) = -1. The left limit and right limit are not equal, hence g(x) is not continuous at x=0.

Determine if |g(x)| is continuous at x=0

Now, let us consider the behavior of |g(x)| around x=0. When x approaches 0 from the right or left, |g(x)| is always 1. The left and right limits exist, are equal, and |g(x)| is defined at x = 0. Hence, |g(x)| is continuous at x=0.

Consider the statement about the continuity of a function and its absolute value

The statement suggests that if the absolute value of a function f, i.e., |f(x)|, is continuous at a point a, the function f itself must also be continuous at that point. To analyze this statement, let's consider the counterexample: Let f(x) = g(x) (the same piecewise function provided in the exercise). As we have seen, |g(x)| is continuous at x=0, but g(x) is not. This counterexample shows that the given statement is false: even if the absolute value of a function is continuous at a point, it does not necessarily imply that the function itself is continuous at the same point.

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by multiple sub-functions, each applicable to a certain part of the function's domain. This means that different equations describe the function's behavior depending on the range of the input value. In simpler terms, think of a piecewise function as a set of rules that tells you which formula to use based on the value of \(x\).

In our example, we have \(g(x) = 1\) if \(x \geq 0\), and \(g(x) = -1\) if \(x < 0\). This means the function \(g(x)\) "switches" the formula at different ranges of \(x\).

Some common features of piecewise functions include:

• They enable modeling of complicated systems or scenarios, such as tax brackets or segments of a journey with different speeds.
• The edges, or points where the sub-functions switch, might present potential discontinuities or jumps, making them crucial points to evaluate the function's behavior.

Understanding the setup of piecewise functions is essential in determining their behavior and continuity at different points.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, regardless of direction. This is always a non-negative value, meaning it strips any minus signs from the numbers. For any real number \(x\):

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

In the context of our function \(g(x)\), the absolute value \(|g(x)|\) takes the same approach:

For the piecewise function \(g(x)\), \(g(x) = 1\) if \(x \geq 0\) and \(g(x) = -1\) if \(x < 0\). Thus, the absolute value \(|g(x)|\) results in \(1\) for both conditions, as \( |1| = 1\) and \( |-1| = 1\).

This process results in \(|g(x)|\) being a constant function with a value of 1 for all \(x\). Understanding how to apply absolute values not only helps simplify functions, but also aids in examining their continuity and limits.

Limits

Limits explore what happens to a function as the input approaches a specific value. They are foundational to understanding the behavior and continuity of functions. In essence, a limit asks: "What does the function approach as \(x\) gets closer to a certain value?"

For the function \(g(x)\), analyzing the limit as \(x\) approaches zero from both sides reveals:

• From the right (\(x \to 0^+\)), \(g(x) = 1\).
• From the left (\(x \to 0^-\)), \(g(x) = -1\).

Since these two limits are not equal, the overall limit at \(x = 0\) does not exist for \(g(x)\), indicating a discontinuity.

However, for the absolute value \(|g(x)|\), both the right and left limits as \(x\) approaches zero yield 1. Consequently, \(|g(x)|\) has a well-defined limit of 1 at 0 and is continuous there. Limits help us diagnose how functions behave near edges and detect potential jumps or holes in their continuity.

Discontinuity

Discontinuity in a function occurs at a point where it is not continuous, meaning there is a break, jump, or hole in the graph of the function. For a function to be continuous at a particular point, the following must be true:

• The function is defined at that point.
• The left-hand limit and right-hand limit at that point exist and are equal.
• The function's value at the point is equal to the limits from either side.

In the given problem, \(g(x)\) is discontinuous at \(x = 0\) because its left and right limits do not match:

• Left limit \(x \to 0^-\) is \(-1\), and right limit \(x \to 0^+\) is 1.

Conversely, the absolute value function \(|g(x)|\) is continuous at the same point. Since \(|g(x)|\) is constantly equal to 1, it meets all conditions for continuity at \(x=0\).

This illustrates an important result: even if the absolute value \(|f(x)|\) of a function \(f(x)\) is continuous at a point, \(f(x)\) itself might not be continuous. Observing discontinuities helps in understanding where and why a function may have breaks or non-uniform behavior and is crucial in calculus and real-world applications.