34.14. Find all differentiable functions $f$ for which $f(x) f^{\prime}(x)=0$ for all $x$.

## 34.5. Functional equations for polynomials
34.14. An equivalent condition is that $\left(f^{2}(x)\right)^{\prime}=0$, hence the function $f^{2}(x)$ is constant. From the continuity of the function $f$, it follows that the function $f(x)$ is also constant.