Let P be the Cantor set. Let f be a bounded real-valued function on [0, 1] which is continuous at every point outside P. Using the definition of Riemann integration, prove that f is Riemann integrable.
Let P be the Cantor set. Let f be a bounded real-valued function on [0, 1] which is continuous at every point outside P. Using the definition of Riemann integration, prove that f is Riemann integrable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Problem Statement**
Let \( P \) be the Cantor set. Let \( f \) be a bounded real-valued function on \([0,1]\) which is continuous at every point outside \( P \). Using the definition of Riemann integration, prove that \( f \) is Riemann integrable.
**Discussion**
The problem involves proving the Riemann integrability of a function \( f \) defined on the interval \([0,1]\). The key condition here is that the function is continuous everywhere except possibly on the Cantor set \( P \). The Cantor set \( P \) is known for being a set of measure zero, which plays a crucial role in determining the integrability of \( f \). By the property of Riemann integration, a bounded function on a closed interval is Riemann integrable if the set of discontinuities has measure zero. Hence, leveraging these properties should guide the proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fad075158-2ba0-404a-8ecf-c1c6a6018f62%2Fwwsgh9e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Let \( P \) be the Cantor set. Let \( f \) be a bounded real-valued function on \([0,1]\) which is continuous at every point outside \( P \). Using the definition of Riemann integration, prove that \( f \) is Riemann integrable.
**Discussion**
The problem involves proving the Riemann integrability of a function \( f \) defined on the interval \([0,1]\). The key condition here is that the function is continuous everywhere except possibly on the Cantor set \( P \). The Cantor set \( P \) is known for being a set of measure zero, which plays a crucial role in determining the integrability of \( f \). By the property of Riemann integration, a bounded function on a closed interval is Riemann integrable if the set of discontinuities has measure zero. Hence, leveraging these properties should guide the proof.
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