(a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). if (x, y) = (0,0),
(a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that g(x, y)| ≤ x|ly| (ii) Using part (i), prove that g(x, y) is continuous at point (0,0). if (x, y) = (0,0),
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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Transcribed Image Text:(a) Assume that g(x, y) is a function of the form
x²+y²
g(x, y)
=
0
if (x, y) = (0,0).
(i) Show that
g(x, y)| ≤ x|ly|
(ii) Using part (i), prove that g(x, y) is continuous at point (0,0).
(b) Show that the function defined by
ay
if (x, y) = (0,0),
x²+y²
f(x, y) =
{}
0
if (x, y) = (0,0)
is not differentiable at point (0,0).
(c) Let z(x, y) = xy be a function defined on a disk D in the positive quadrant
containing the point (1,2). Prove whether z(x, y) satisfies the Clairaut Theorem
at (1,2).
if (x, y) = (0,0),
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