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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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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