Consider the function f : R^2 → R defined by f(x, y) = {(cos x sin y)/y if y ̸= 0 cos x if y = 0. Is f continuous everywhere? Justify your answer.
Consider the function f : R^2 → R defined by f(x, y) = {(cos x sin y)/y if y ̸= 0 cos x if y = 0. Is f continuous everywhere? Justify your answer.
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.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 37CR
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Question
Consider the function f : R^2 → R defined by
f(x, y) =
{(cos x sin y)/y
if y ̸= 0
cos x if y = 0.
Is f continuous everywhere? Justify your answer.
(b) Let g : R
2 → R be defined by
g(x, y) =
{2xy*(x^2-y^2))/x^2+y^2 if (x, y) ̸= (0, 0)
a if (x, y) = (0, 0).
(i) For what value(s) of a, if any, is g continuous at (0, 0)?
(ii) For these value(s) of a, show that gx(0, y) = −2y for any y ∈ R
and gy(x, 0) = 2x for any x ∈ R.
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