Find if the function is Continuous and whether the partial derivatives exist

Question

$$f(x,y)=\begin{Bmatrix} 0 & (x,y)=(0,0)\\ \frac{xy}{x^{2}+y^{2}}&(x,y)\neq (0,0) \end{Bmatrix}$$

How to find out if this is continuous?

And do the partial derivatives exist at $(0,0)$?

Answer 1

hint

For $x\ne 0$

$$f (x,x)=\frac {1}{2} $$

$$f (x,2x) =\frac {2}{5} .$$

What hapens when $x\to 0$.

Answer 2

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 $${ f }_{ x }^{ \prime }\left( 0,0 \right) =\lim _{ x\rightarrow 0 }{ \frac { f\left( x,0 \right) -f\left( 0,0 \right) }{ x } } =\lim _{ x\rightarrow 0 }{ \frac { \frac { x\cdot 0 }{ { x }^{ 2 }+{ 0 }^{ 2 } } -0 }{ x } } =0\\ { f }_{ y }^{ \prime }\left( 0,0 \right) =\lim _{ x\rightarrow 0 }{ \frac { f\left( 0,y \right) -f\left( 0,0 \right) }{ y } } =\lim _{ y\rightarrow 0 }{ \frac { \frac { 0\cdot y }{ { 0 }^{ 2 }+{ y }^{ 2 } } -0 }{ y } } =0\\ $$