2. Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a continuous function and $M=\left\{\left.A(x)=\left(\begin{array}{cc}x & f(x) \\ 0 & x\end{array}\right) \right\rvert\, x \in(0, \infty)\right\}$. Determine the functions $f$ for which $M$ is a group with respect to matrix multiplication.
2. It is observed that $A(x) \cdot A(y)=\left(\begin{array}{cc}x y & x f(y)+y f(x) \\ 0 & x y\end{array}\right)$. Since $A(x) \cdot A(y)=A(x y)$, it follows that $f(x y)=x \cdot f(y)+y \cdot f(x), \forall x \in(0, \infty)$.

With the notation $g(x)=\frac{f(x)}{x}, x>0$, we obtain that the continuous function $g$ satisfies the functional equation $g(x y)=g(x)+g(y), \forall x \in(0, \infty)$. Then $g(x)=k \log _{a} x$, hence $f(x)=k x \log _{a} x$, $\forall x \in(0, \infty)$, where $k \in \mathbb{R}, a \in(0,1) \cup(1, \infty)$.

For functions $f$ of this form, the group axioms are immediately verified.