Definition: A function f : R → R is periodic if there exists k > 0 such that f(x + k) = f(x) for all r €R (e.g., sin(r + 27) = sin r). If f : R → R is continuous and periodic, prove that f is uniformly continuous. Hint: Show that f is uniformly continuous on [-k, k] and then use the definition of uniformly continuous to show that f is uniformly continuous on R.
Definition: A function f : R → R is periodic if there exists k > 0 such that f(x + k) = f(x) for all r €R (e.g., sin(r + 27) = sin r). If f : R → R is continuous and periodic, prove that f is uniformly continuous. Hint: Show that f is uniformly continuous on [-k, k] and then use the definition of uniformly continuous to show that f is uniformly continuous on R.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Definition: A function f : R → R is periodic if there exists k > 0 such that
f(x + k) = f(x) for all r €R (e.g., sin(r + 27) = sin r).
If f : R → R is continuous and periodic, prove that f is uniformly continuous.
Hint: Show that f is uniformly continuous on [-k, k] and then use the definition of
uniformly continuous to show that f is uniformly continuous on R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5a7bb65-b92b-4b2a-a8a3-c52d52b3b98d%2F39cb06d4-41a7-445f-be86-6b32dc5603ac%2Fv70mcn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Definition: A function f : R → R is periodic if there exists k > 0 such that
f(x + k) = f(x) for all r €R (e.g., sin(r + 27) = sin r).
If f : R → R is continuous and periodic, prove that f is uniformly continuous.
Hint: Show that f is uniformly continuous on [-k, k] and then use the definition of
uniformly continuous to show that f is uniformly continuous on R.
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