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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![### Question
Is the following function continuous on \((-\infty, \infty)\)?
\[
f(x) =
\begin{cases}
\sin x & \text{if } x < \pi/4 \\
\cos x & \text{if } x \geq \pi/4
\end{cases}
\]
### Explanation
This is a piecewise function where:
- \(f(x) = \sin x\) is applied when \(x\) is less than \(\pi/4\).
- \(f(x) = \cos x\) is applied when \(x\) is greater than or equal to \(\pi/4\).
To determine if the function is continuous on the entire real number line, we need to check the continuity at the point \(x = \pi/4\) where the function definition changes. The function is continuous where \(\sin x\) and \(\cos x\) individually are continuous, but special attention is needed at \(x = \pi/4\) due to the transition between the two pieces. For the function to be continuous at \(x = \pi/4\), both the left-hand limit and right-hand limit as \(x\) approaches \(\pi/4\) must equal \(f(\pi/4)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f1c68a1-c113-41cc-b0cf-42f2666e5687%2F2ba2855f-7565-49a2-8c9f-673800968832%2F9jojlia_processed.png&w=3840&q=75)
Transcribed Image Text:### Question
Is the following function continuous on \((-\infty, \infty)\)?
\[
f(x) =
\begin{cases}
\sin x & \text{if } x < \pi/4 \\
\cos x & \text{if } x \geq \pi/4
\end{cases}
\]
### Explanation
This is a piecewise function where:
- \(f(x) = \sin x\) is applied when \(x\) is less than \(\pi/4\).
- \(f(x) = \cos x\) is applied when \(x\) is greater than or equal to \(\pi/4\).
To determine if the function is continuous on the entire real number line, we need to check the continuity at the point \(x = \pi/4\) where the function definition changes. The function is continuous where \(\sin x\) and \(\cos x\) individually are continuous, but special attention is needed at \(x = \pi/4\) due to the transition between the two pieces. For the function to be continuous at \(x = \pi/4\), both the left-hand limit and right-hand limit as \(x\) approaches \(\pi/4\) must equal \(f(\pi/4)\).
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