If f : R + Ris continuous and periodic, prove that f is uniformly continuous.
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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I need help with #5.
![1. Use the definition to prove that the function in f(x) = 2r – 3 is uniformly continuous.
2. Use the definition to prove f = is uniformly continuous on [0, 00).
3. Suppose f, g : D → R are uniformly continuous on D. Prove that f + g is uniformly
continuous on D.
4. Let f : X → Y and g : Y → Z be uniformly continuous on X and Y, respectively.
Prove gof : X → Z is uniformly continuous on X.
5. Definition: A function f : R → R is periodic if there exists k > 0 such that
f(r + k) = f(x) for all z € R (e.g., sin(r + 2ñ) = sin x).
If f : R → R is continuous and periodic, prove that f is uniformly continuous.
Hint: Show that ƒ 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%2F652eda4a-07c7-446c-baaa-e1ffddd9731f%2Fe9baf7dd-cd94-4450-81f4-5d7562cd2388%2Frx5jiu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Use the definition to prove that the function in f(x) = 2r – 3 is uniformly continuous.
2. Use the definition to prove f = is uniformly continuous on [0, 00).
3. Suppose f, g : D → R are uniformly continuous on D. Prove that f + g is uniformly
continuous on D.
4. Let f : X → Y and g : Y → Z be uniformly continuous on X and Y, respectively.
Prove gof : X → Z is uniformly continuous on X.
5. Definition: A function f : R → R is periodic if there exists k > 0 such that
f(r + k) = f(x) for all z € R (e.g., sin(r + 2ñ) = sin x).
If f : R → R is continuous and periodic, prove that f is uniformly continuous.
Hint: Show that ƒ 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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