Let $\mathrm{f}$ be a complex-valued function with the following property:
$$\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w) $$
for all $w,z \in \mathbb C$. Necessary conditions are that $\mathrm{f}(0)=\mathrm{f}(\pm 1) = \mathrm{f}(\pm\mathrm{i})=0$.
One obvious example is the zero function: $\mathrm{f}(z)=0$ for all $z \in \mathbb C$.
- Are there any other examples of functions which satisfy the above condition?
- If so, what are the "nicest" examples, e.g. continuous, holomorphic, biholomorphic?