Show that there is no injection from the power-set of $\mathbb{R}$ to $\mathbb{R}$. Show also that there is an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

Let $X$ be the set of all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)=x$ for all but finitely many $x$. Determine whether or not there exists an injection from $X$ to $\mathbb{R}$.