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jump discontinulity at 9, removable discontinuity at 7 y 9. 9,

jump discontinulity at 9, removable discontinuity at 7 y 9. 9,

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Understanding Discontinuities in Functions This page presents a task that involves sketching graphs of a function \( f \) defined on the real numbers \( \mathbb{R} \), which is continuous except at specified points of discontinuity. #### Key Discontinuities: - **Jump Discontinuity at \( x = 9 \)** - **Removable Discontinuity at \( x = 7 \)** ### Graph Analysis: 1. **Diagram (a):** - The graph is smooth with no breaks except at two points. - At \( x = 7 \), there is a removable discontinuity indicated by a hole in the graph. - At \( x = 9 \), there is a jump discontinuity where the graph abruptly shifts to a different y-value. 2. **Diagram (b):** - The function starts on one level and, upon reaching \( x = 7 \), there is a gap (removable discontinuity). - As it approaches \( x = 9 \), the graph jumps to a different y-value, indicating a jump discontinuity. 3. **Diagram (c):** - Shows a continuous curve interrupted at \( x = 7 \) with a hole marking a removable discontinuity. - There is a visible jump at \( x = 9 \), with the subsequent part of the graph on a different level. 4. **Diagram (d):** - The graph displays a continuous path, except for the gap at \( x = 7 \). - At \( x = 9 \), there is a jump in the y-value, representing the jump discontinuity. ### Explanation of Discontinuities: - **Removable Discontinuity:** It occurs at \( x = 7 \), where the graph has an isolated hole. The function can be redefined or extended to remove the discontinuity. - **Jump Discontinuity:** Observed at \( x = 9 \), where there is an abrupt change in the function's value, causing a break in the graph. This type of discontinuity cannot be eliminated by simply redefining the function at the point. These diagrams help illustrate how specific discontinuities affect the behavior of a function across its domain.
Transcribed Image Text:### Understanding Discontinuities in Functions This page presents a task that involves sketching graphs of a function \( f \) defined on the real numbers \( \mathbb{R} \), which is continuous except at specified points of discontinuity. #### Key Discontinuities: - **Jump Discontinuity at \( x = 9 \)** - **Removable Discontinuity at \( x = 7 \)** ### Graph Analysis: 1. **Diagram (a):** - The graph is smooth with no breaks except at two points. - At \( x = 7 \), there is a removable discontinuity indicated by a hole in the graph. - At \( x = 9 \), there is a jump discontinuity where the graph abruptly shifts to a different y-value. 2. **Diagram (b):** - The function starts on one level and, upon reaching \( x = 7 \), there is a gap (removable discontinuity). - As it approaches \( x = 9 \), the graph jumps to a different y-value, indicating a jump discontinuity. 3. **Diagram (c):** - Shows a continuous curve interrupted at \( x = 7 \) with a hole marking a removable discontinuity. - There is a visible jump at \( x = 9 \), with the subsequent part of the graph on a different level. 4. **Diagram (d):** - The graph displays a continuous path, except for the gap at \( x = 7 \). - At \( x = 9 \), there is a jump in the y-value, representing the jump discontinuity. ### Explanation of Discontinuities: - **Removable Discontinuity:** It occurs at \( x = 7 \), where the graph has an isolated hole. The function can be redefined or extended to remove the discontinuity. - **Jump Discontinuity:** Observed at \( x = 9 \), where there is an abrupt change in the function's value, causing a break in the graph. This type of discontinuity cannot be eliminated by simply redefining the function at the point. These diagrams help illustrate how specific discontinuities affect the behavior of a function across its domain.
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