We equip $\mathscr{M}_{d}(\mathbb{R})$ with the topology associated with the norm $\|M\| = \sqrt{\langle M, M \rangle}$.
\begin{itemize}
\item[(a)] Show that the map $f : \mathscr{M}_{d}(\mathbb{R}) \rightarrow \mathscr{M}_{d}(\mathbb{R})$ defined by $f(M) = M^{T}M$ is continuous.
\item[(b)] Show that $\mathrm{SO}_{d}(\mathbb{R})$ is a closed bounded subset of $\mathscr{M}_{d}(\mathbb{R})$.
\end{itemize}