Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

What points, if any, are the functions discontinuous? $$ g(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x<0 \\ -x & \text { if } 0 \leq x \leq 1 \\ x & \text { if } x>1 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function \( g(x) \) is piecewise, meaning it is defined differently over different intervals of \( x \). It has three branches: \( x^2 \) for \( x < 0 \), \( -x \) for \( 0 \leq x \leq 1 \), and \( x \) for \( x > 1 \). To check for discontinuities, we need to examine these transition points at \( x = 0 \) and \( x = 1 \).

Step 2: Check Continuity at \( x = 0 \)

Evaluate the left-hand limit as \( x \) approaches 0 from the left: \[ \lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} x^2 = 0^2 = 0. \]Evaluate the right-hand limit as \( x \) approaches 0 from the right:\[ \lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} -x = -0 = 0. \]Also, find \( g(0) \):\[ g(0) = -0 = 0. \]Since the left-hand limit, right-hand limit, and \( g(0) \) are all equal to 0, the function is continuous at \( x = 0 \).

Step 3: Check Continuity at \( x = 1 \)

Evaluate the left-hand limit as \( x \) approaches 1 from the left: \[ \lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} -x = -1. \]Evaluate the right-hand limit as \( x \) approaches 1 from the right:\[ \lim_{x \to 1^+} g(x) = \lim_{x \to 1^+} x = 1. \]Find \( g(1) \):\[ g(1) = -1. \]The left-hand limit and \( g(1) \) are both \(-1\), but the right-hand limit is \(1\). Since the limits from the left and right do not match, the function is discontinuous at \( x = 1 \).

Step 4: Conclusion on Discontinuity

The function \( g(x) \) is continuous at \( x = 0 \), as all limits and the function value agree. However, it is discontinuous at \( x = 1 \) because the limits from each side aren't equal.

Key Concepts

Continuity and Discontinuity

When studying piecewise functions like the one given in the exercise, the concepts of continuity and discontinuity come into play quite often. A function is said to be continuous at a point if the graph of the function has no breaks, jumps, or holes at that point. This happens when the left-hand limit and the right-hand limit at a point are equal, and they also equal the function's value at that point. In our exercise, we examined the function at transition points, specifically at \(x = 0\) and \(x = 1\).

• At \(x = 0\), both the left-hand limit \( \lim_{x \to 0^-} g(x) \), the right-hand limit \( \lim_{x \to 0^+} g(x) \), and the value \( g(0) \) all equal 0. Therefore, the function is continuous at \(x = 0\).
• At \(x = 1\), the left-hand limit \( \lim_{x \to 1^-} g(x) = -1\), and the right-hand limit \( \lim_{x \to 1^+} g(x) = 1\) do not match. This shows an abrupt change or jump at \(x = 1\), making the function discontinuous at this point.

Determining the points of discontinuity helps us understand the behavior of the function across its domain, which is crucial in both graphing and applying further calculus concepts.

Limits

Limits are a foundational concept in calculus and play a vital role in determining the continuity of functions. In our piecewise function, evaluating limits at respective points of interest is crucial to identify whether there is continuity or a possible discontinuity. The idea of a limit is to find out what value a function approaches as the input (or \(x\)-value) gets arbitrarily close to a certain point.

• Left-hand limit (\( \lim_{x \to c^-} f(x) \)): This limit represents how the function behaves as \(x\) approaches a specific point \(c\) from the left.
• Right-hand limit (\( \lim_{x \to c^+} f(x) \)): Conversely, this is how the function behaves as \(x\) approaches \(c\) from the right.

For a function to be continuous at a point, both these limits must exist, be equal, and also equal the value of the function at that point. If any of these conditions fail, a discontinuity exists.

Functions and Graphs

Understanding the behavior of piecewise functions can be best visualized through their graphs. Graphs provide a visual means to comprehend how different segments of the piecewise function connect or disconnect at their transition points. For our piecewise function, the graph would consist of three distinct parts:

• A parabolic section \(x^2\) when \(x < 0\).
• A linear section descending with slope \(-1\) for \(0 \leq x \leq 1\).
• Another linear section ascending with slope \(1\) for \(x > 1\).

It is essential to carefully plot these sections and observe their connections at the boundaries. At \(x = 0\), you would see a smooth transition, confirming continuity. However, at \(x = 1\), the gap between the descent from \(-1\) to the rise at \(1\) will be clear on the graph due to this point's discontinuity. Graphs not only show us continuity and discontinuity but also help us build a deeper understanding of function behavior.

Calculus Education

Calculus education focuses on understanding the language of functions, limits, derivatives, and integrals. These foundational concepts are interconnected and provide the tools needed for deeper mathematical exploration. Piecewise functions serve as a perfect educational tool because they embody real-world situations where systems behave differently under varying conditions. To approach calculus effectively:

• Grasp Basic Functionality: Start with an understanding of different types of functions, including linear, quadratic, and piecewise.
• Master Limits: Learn how to find and understand both left-hand and right-hand limits effectively.
• Bridge with Continuity: Use limits to assess continuity at points, an essential step before learning differentiation.

Understanding these concepts aids students in tackling more advanced calculus problems, including those involving derivatives and integrals, ultimately making calculus more approachable and less daunting.