Consider the map $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 } , ( x , y ) \mapsto \left( - x - y ^ { 2 } , y + x ^ { 2 } \right)$. Pick the correct statement(s) from below. (A) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that there is an open neighbourhood $U$ of $( a , b )$ such that $\left. f \right| _ { U }$ is a homeomorphism from $U$ to $f ( U )$. (B) The derivative $D f$ maps some non-zero tangent vector to $\mathbb { R } ^ { 2 }$ at $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, to the zero tangent vector at $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$. (C) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that for every open neighbourhood $U$ of $( a , b ) , \left. f \right| _ { U }$ is not a homeomorphism from $U$ to $f ( U )$. (D) For every differentiable curve $\gamma$ through $( 0,0 ) , f \circ \gamma$ is differentiable curve.
Consider the map $f : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 } , ( x , y ) \mapsto \left( - x - y ^ { 2 } , y + x ^ { 2 } \right)$. Pick the correct statement(s) from below.\\
(A) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that there is an open neighbourhood $U$ of $( a , b )$ such that $\left. f \right| _ { U }$ is a homeomorphism from $U$ to $f ( U )$.\\
(B) The derivative $D f$ maps some non-zero tangent vector to $\mathbb { R } ^ { 2 }$ at $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, to the zero tangent vector at $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$.\\
(C) There exist infinitely many $( a , b ) \in \mathbb { R } ^ { 2 }$ such that for every open neighbourhood $U$ of $( a , b ) , \left. f \right| _ { U }$ is not a homeomorphism from $U$ to $f ( U )$.\\
(D) For every differentiable curve $\gamma$ through $( 0,0 ) , f \circ \gamma$ is differentiable curve.