Q2 Let f(x) = x² for æ rational and f(x) = 0 for x irrational. (a) Prove f is continuous at x = 0. (b) Prove f is discontinuous at all x 0. (c) Prove f is differentiable at x = 0. (Warning: You cannot simply claim f'(x) = 2x.)
Q2 Let f(x) = x² for æ rational and f(x) = 0 for x irrational. (a) Prove f is continuous at x = 0. (b) Prove f is discontinuous at all x 0. (c) Prove f is differentiable at x = 0. (Warning: You cannot simply claim f'(x) = 2x.)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.2: Partial Derivatives
Problem 48E
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![Q2
Let f(x) = x² for æ rational and f(x) = 0 for x irrational.
(a) Prove f is continuous at x = 0.
(b) Prove f is discontinuous at all x 0.
(c) Prove f is differentiable at x = 0. (Warning: You cannot simply claim f'(x) = 2x.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F655e5d67-ab19-404c-b883-3aa3c693f6a1%2F80d720b6-0c95-4356-9dd1-5ec493bf1909%2F5r9d7bm_processed.png&w=3840&q=75)
Transcribed Image Text:Q2
Let f(x) = x² for æ rational and f(x) = 0 for x irrational.
(a) Prove f is continuous at x = 0.
(b) Prove f is discontinuous at all x 0.
(c) Prove f is differentiable at x = 0. (Warning: You cannot simply claim f'(x) = 2x.)
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