The general solution of the exponential Cauchy equation on a bounded restricted domain

Summary.

The equation

$$ f(x + y) = f(x)f(y), \quad (*) $$

for real functions of a real variable, is studied in this paper on a triangular restricted domain in ${\mathbb{R}^2}$.

Main results: 1) In the class of nowhere vanishing functions, the general local solutions are restrictions of solutions of other suitable equations on the whole space, more general than (*). A related extension theorem is proved.

2) If f vanishes in some points of its domain, the above-mentioned behaviour fails to hold true and the general local solution consists of a function defined by means of identically zero and arbitrary functions.