Few functions and I have to study the following aspects: Continuity in the point (0,0), If the derivative exists at (0,0), Continuity of the partial derivatives at (0,0), Directional derivatives at (0,0)

Elise Kelley 2022-10-25 Answered
Few functions and I have to study the following aspects:
Continuity in the point (0,0)
If the derivative exists at (0,0)
Continuity of the partial derivatives at (0,0)
Directional derivatives at (0,0)
One of the functions is, for example:
f ( x , y ) = { x 2 y 2 ( x 2 + y 2 ) , if if (x,y) not (0,0) 0 , if (x,y) = (0,0)
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Answers (1)

Laci Conrad
Answered 2022-10-26 Author has 17 answers
Use the polar coordinates transformation:
{ x = r cos θ y = r sin θ
Then, f ( x , y ) = r 3 cos 2 θ sin 2 θ
Now, since cos 2 θ sin 2 θ is bounded, one can use the squeeze theorem.
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