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Is this function class $C^1$ at $(0,0)$ ? $$\left\{\begin{matrix} \frac{2x^2y+y^3}{x^2+y^2} &(x,y)\neq0 \\ \\ 0& (x,y)=0 \end{matrix}\right.$$ I thought like this, first i took partial derivative with respect to x and y: $$\frac{\partial f}{\partial x}=2y \quad \frac{\partial f}{\partial y}=\frac{2x^2+3y^2}{2y}$$and beacause $\frac{\partial f}{\partial y}=\frac{2x^2+3y^2}{2y}$ is not defined at zero i made conclusion that this function is not class $C^1$

Is that type of reasoning correct?

Thanks in advance

Arctic Char
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simon
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1 Answers1

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We need to show that the funtion has continuous partial derivatives at $(0,0)$ with

$$f_x=\frac{2 x y^3}{(x^2 + y^2)^2} \quad f_y = \frac{2 x^4 + x^2 y^2 + y^4}{(x^2 + y^2)^2}$$

and we can easily check that they are not continuous at $(0,0)$.

Refer also to

user
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