4. A theorem states that any continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b] must be bounded, and must have a max and a min on [a, b]. • Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous on (0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1,2] in the notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly continuous on any interval [a, b]. • Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of f(x) = a² is not uniformly continuous on [0, 0) in the notes. • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b). Does this contradict the theorem stated above?
4. A theorem states that any continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b] must be bounded, and must have a max and a min on [a, b]. • Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous on (0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1,2] in the notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly continuous on any interval [a, b]. • Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of f(x) = a² is not uniformly continuous on [0, 0) in the notes. • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b). Does this contradict the theorem stated above?
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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#4. First 3 bullet points. Thanks.
![4. A theorem states that any continuous function on a closed and bounded interval [a, b] is
uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b]
must be bounded, and must have a max and a min on [a, b].
• Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous
on [0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1, 2] in the
notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly
continuous on any interval [a, b].
• Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of
f(x) = x² is not uniformly continuous on [0, 0) in the notes.
• Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum
on (a, b). Does this contradict the theorem stated above?
Give an example of a function f(x) that is not bounded on [a, b). Does this contradict
the theorem stated above?
• What is the max and the min of f(x) = x³ on [1,3)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F1e6a9f56-3c4d-471d-aeb1-7fc12189d2d8%2Fla4lxkt_processed.png&w=3840&q=75)
Transcribed Image Text:4. A theorem states that any continuous function on a closed and bounded interval [a, b] is
uniformly continuous on [a, b]. Another theorem states that a continuous function on [a, b]
must be bounded, and must have a max and a min on [a, b].
• Use the definition of uniform continuity to show that f(x) = x³ is uniformly continuous
on [0, 1]. Hint: Emulate the proof of f(x) = x² is uniformly continuous on [1, 2] in the
notes. Notice that the same proof can be used to show that f(x) = x³ is uniformly
continuous on any interval [a, b].
• Show that f(x) = x³ is not uniformly continuous on [0, 0). Hint: Emulate the proof of
f(x) = x² is not uniformly continuous on [0, 0) in the notes.
• Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum
on (a, b). Does this contradict the theorem stated above?
Give an example of a function f(x) that is not bounded on [a, b). Does this contradict
the theorem stated above?
• What is the max and the min of f(x) = x³ on [1,3)?
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