The problem is this:
Let the matric $J$ be$$ J = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$$
Let $f : \Omega \rightarrow \mathbb{R}^2$be an $\mathbb{R}$- differentiable function defined on $\Omega \subset \mathbb{R}^2$.
a) Show that $f \in O(\Omega)$ if and only if for any $(x, y) \in \Omega$ we have $J \circ d_{(x,y)}f = d_{(x,y)}f \circ J$.
b) Show that $f \in O(\Omega)$ if and only if for any $(x, y) \in \Omega$ and any $\theta \in \mathbb{R}$ we have $R_{\theta} \circ d_{(x, y)}f = d_{(x, y)}f \circ R_{\theta}$ where $R_{\theta} : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ denotes the rotation of angle $\theta$.
c) Let $\theta \in \mathbb{R}$. Assume that $R_{\theta} \circ d_{(x, y)}f = d_{(x, y)}f \circ R_{\theta}$ for any $(x, y) \in \Omega$. Is $f \in O(\Omega)$?
I don't know how to even start with this and i'm not sure I understood the questions with the notation used..
Any help would be much appreciated