<s>[INST] Find if the function is Continuous and whether the partial derivatives exist 
$$f(x,y)=\begin{Bmatrix}
0 & (x,y)=(0,0)\\ 
 \frac{xy}{x^{2}+y^{2}}&(x,y)\neq (0,0) 
\end{Bmatrix}$$

How to find out if this is continuous?
And do the partial derivatives exist at $(0,0)$?
 [/INST] Note that it is not differential at $(0,0)$ $$\left( \frac { 1 }{ n } ,\frac { 1 }{ n }  \right) \overset { n\rightarrow \infty  }{ \longrightarrow  } \left( 0,0 \right) ,\left( \frac { 2 }{ n } ,\frac { 1 }{ n }  \right) \overset { n\rightarrow \infty  }{ \longrightarrow  } \left( 0,0 \right) \\ f\left( \frac { 1 }{ n } ,\frac { 1 }{ n }  \right) =\frac { \frac { 1 }{ { n }^{ 2 } }  }{ \frac { 2 }{ { n }^{ 2 } }  } \longrightarrow \frac { 1 }{ 2 } \\ f\left( \frac { 2 }{ n } ,\frac { 1 }{ n }  \right) =\frac { \frac { 2 }{ { n }^{ 2 } }  }{ \frac { 5 }{ { n }^{ 2 } }  } \longrightarrow \frac { 2 }{ 5 } \\ \quad $$
And partial derivatives are $${ f }_{ x }^{ \prime  }\left( 0,0 \right) =\lim _{ x\rightarrow 0 }{ \frac { f\left( x,0 \right) -f\left( 0,0 \right)  }{ x }  } =\lim _{ x\rightarrow 0 }{ \frac { \frac { x\cdot 0 }{ { x }^{ 2 }+{ 0 }^{ 2 } } -0 }{ x }  } =0\\ { f }_{ y }^{ \prime  }\left( 0,0 \right) =\lim _{ x\rightarrow 0 }{ \frac { f\left( 0,y \right) -f\left( 0,0 \right)  }{ y }  } =\lim _{ y\rightarrow 0 }{ \frac { \frac { 0\cdot y }{ { 0 }^{ 2 }+{ y }^{ 2 } } -0 }{ y }  } =0\\  $$
</s>