math237_a3 - SOLUTIONS FOR ASSIGNMENT 3 Problem Set 2, A6:...

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SOLUTIONS FOR ASSIGNMENT 3 Problem Set 2, A6: Let f ( x, y ) = x | y 1 | . Use the definition to prove that f is not differentiable at (1 , 1) . Note that the point (1 , 1) is a typical point as regards differentiation w.r.t x but is an exceptional point as regards differentiation w.r.t. y . Since ∂f ∂x = | y 1 | , it follows that ∂f ∂x (1 , 1) = 0 . Consider now lim h 0 f (1 , 1 + h ) f (1 , 1) h = lim h 0 | h | h = +1 , if h > 0 = 1 , if h < 0 . The above limit does not exist and thus ∂f ∂y (1 , 1) does not exist either. This im- plies that f is not differentiable at (1 , 1) , by the definition of differentiability. Problem Set 2, B7(i): Conside the function f ( x, y ) = ( xy ) 2 / 3 . a). Use the definition to determine whether f is differentiable at (0 , 0) . b). Using the answer to part a), can you use one of the theoretical results to draw a conclusion concerning the continuity of f at (0 , 0) ? c). Using the answer to part a), can you use one of the theoretical results to draw a conclusion concerning the continuity of f x and f y at (0 , 0) . a). Note that f is symmetric in x and y and that (0 , 0) is an exceptional point as regards differentiation w.r.t both x and y . We need to use the definition of dif- ferentiability to check whether f is differentiable at (0 , 0) . Consider
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math237_a3 - SOLUTIONS FOR ASSIGNMENT 3 Problem Set 2, A6:...

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