I am interested in generalisation of the following fact [known as automatic continuity, as I have been pointed out below]. I am especially looking for references to papers dating back to 1920s---I feel that question like these have been well-studied when people have still been interested in set-theoretic aspects of analysis...

(Cauchy) Any measurable automorphism (R,+)-->(R,+) is necessary a linear function, and any measurable homomorphism $(R,+)\longrightarrow(R,\times)$ is necessary an exponential function $e^{ax}$. Is something similar true for homomophisms of complex numbers f:(C,+)--->(C,+) or f:(C,+)-->(C,*) (latter assuming ker f = Z) ? (Yes, see answers below).

That is, I am interested in facts which follow the following rough pattern:

If a map is not set-theoretically wild, e.g. measurable or Borel, and satisfies some identities, then it is in fact continuous, and, further, can be given by an explicit formula.