Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ g(x)= \begin{cases}x^{2} & \text { if } x<0 \\ -x & \text { if } 0 \leq x \leq 1 \\ x & \text { if } x>1\end{cases} $$

Step-by-Step Solution

Answer

The function is discontinuous at \( x = 1 \).

Step 1: Identify Domains for Each Case

Examine the piecewise function to determine the domain for each expression. The function is divided into three cases: 1. \( g(x) = x^2 \) for \( x < 0 \)2. \( g(x) = -x \) for \( 0 \leq x \leq 1 \)3. \( g(x) = x \) for \( x > 1 \)

Step 2: Check Continuity at the Transition Points

Verify continuity at the transition points, which are the points where the function changes its expression form. The critical points to check are \( x = 0 \) and \( x = 1 \).

Step 3: Check Continuity at x = 0

To check continuity at \( x = 0 \):- Left-hand limit: \( \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} x^2 = 0 \).- Right-hand limit: \( \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} -x = 0 \).- Value of function: \( g(0) = -0 = 0 \).Since the left-hand limit, right-hand limit, and the function value are equal, the function is continuous at \( x = 0 \).

Step 4: Check Continuity at x = 1

To check continuity at \( x = 1 \):- Left-hand limit: \( \lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} -x = -1 \).- Right-hand limit: \( \lim_{x \to 1^+} g(x) = \lim_{x \to 1^+} x = 1 \).- Value of function: \( g(1) = -1 \).The left-hand limit \( -1 \) does not equal the right-hand limit \( 1 \), hence the function is discontinuous at \( x = 1 \).

Step 5: Conclusion

Evaluate the function and ensure all transition points have been checked. The function is continuous for all \( x \) except at \( x = 1 \) where a discontinuity occurs due to differing one-sided limits.

Key Concepts

Continuity

Continuity in functions is all about smoothness—no breaks, jumps, or abrupt changes in the graph of the function. A function is said to be continuous at a point if it meets three essential criteria at that point:

• The function is defined at that point.
• The limit of the function as it approaches the point is equal from both sides (left and right).
• The value of the function at that point is equal to that common limit.

In the given piecewise function, we explore continuity at specific points where the function expression changes, known as transition points. A function like this is continuous over segments defined by the conditions in its piecewise setup, as long as it meets the continuity criteria at the transition points.

Discontinuity

A function is discontinuous at a point if one or more criteria for continuity are not met. Discontinuities can come in various types, like jump discontinuities, removable discontinuities, and infinite discontinuities, each depicted by different behaviors.

• Jump Discontinuity: This occurs when the left and right-hand limits exist but are not equal. This is visible as a sudden 'jump' in the graph of the function.
• Removable Discontinuity: Occurs when a point limit exists, but it differs from the function's value.
• Infinite Discontinuity: Happens when one or both of the one-sided limits go to infinity.

In our piecewise function example, at the point \( x = 1 \), the function displays a jump discontinuity. The left-hand limit is \(-1\) while the right-hand limit is \(1\), indicating a sudden change, demonstrating the nature of jump discontinuity.

Limits

Limits are essential in understanding the behavior of functions at specific points, especially when dealing with piecewise functions. They help us make sense of what the function is approaching as the input values get closer from either side of a particular point.

• Left-Hand Limit (LHL): Represents the value that the function approaches as the input comes from the left side of the point.
• Right-Hand Limit (RHL): Represents the value that the function approaches from the right side of the point.
• Two-sided Limit: If the LHL and RHL agree, this common value is the actual limit at that point.

In our function, we examine the limits at \( x = 0 \) and \( x = 1 \). At \( x = 0 \), both one-sided limits and the function's value meet at \( 0 \). Conversely, at \( x = 1 \), the limits diverge \((-1 \) and \(1\)). These limits are critical in determining the nature of continuity at these points.

Transition Points

Transition points in a piecewise function are the x-values where the function switches from one rule to another. These points are crucial because they are the potential spots for discontinuities, making them the primary focus when checking for continuity.
To evaluate continuity at these transition points, we consider:

• The left-hand and right-hand limits.
• The actual function value at the point itself.

In the exercise, the transition points are \( x = 0 \) and \( x = 1 \). At \( x = 0 \), the function smoothly transitions as both the limits and the function value agree, ensuring continuity. However, at \( x = 1 \), the mismatch between the limits indicates a discontinuity. Understanding these transition points is key to identifying where the function might not be continuous.