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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.)

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
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.)
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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