34.9. Find all continuous functions that are defined for all $x$ and satisfy the relation $f(x)=$ $=a^{x} f(x / 2)$, where $a$ is a fixed positive number.
34.9. It is clear that $f(x)=a^{x} f(x / 2)=a^{x} a^{x / 2} f(x / 4)=a^{x} a^{x / 2} a^{x / 4} f(x / 8)=$ $=\ldots=a^{x\left(1+1 / 2+1 / 4+\ldots+1 / 2^{k}\right)} f\left(x / 2^{k+1}\right)$. If $k \rightarrow \infty$, then $f\left(x / 2^{k+1}\right) \rightarrow f(0)$, since the function $f$ is continuous. Moreover, $1+\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^{k}} \rightarrow 2$ as $k \rightarrow \infty$. Therefore, $f(x)=a^{2 x} f(0)$. Thus, $f(x)=C a^{2 x}$, where $C$ is an arbitrary constant.