Let $u$ and $v$ be real-valued functions on $\mathbb { R } ^ { 2 }$ defined as follows:
$$\begin{aligned} & u ( x , y ) = \begin{cases} \frac { x ^ { 3 } - 3 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \\ & v ( x , y ) = \begin{cases} \frac { y ^ { 3 } - 3 y x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } & \text { if } ( x , y ) \neq ( 0,0 ) \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$
Let $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$ be the function $f ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. Pick the correct statement(s) from below.
(A) $\frac { \partial u } { \partial x } , \frac { \partial u } { \partial y } , \frac { \partial v } { \partial x }$ and $\frac { \partial v } { \partial y }$ exist at $( 0,0 )$.
(B) $\frac { \partial u } { \partial x }$ is continuous at $( 0,0 )$.
(C) For every fixed $( a , b ) \neq ( 0,0 ) \in \mathbb { R } ^ { 2 }$, the function $t \mapsto f ( t a , t b )$ is a differentiable function (of $t$ ).
(D) $f$ is differentiable at $( 0,0 )$.