The general solution of the exponential Cauchy equation on a bounded restricted domain (Q2369842)

The exponential functional equation \[ f(x+y)=f(x)f(y) \] is studied for functions \(f:E(a,b;r)\to{\mathbb R}\) defined on a triangle \[ E(a,b;r):=\bigl\{(x,y)\in{\mathbb R}^2:x\geq a,\;y\geq b,\;x+y<a+b+r\bigr\}. \] To give the general solution two cases are needed to consider. 1. If \(f\neq 0\) on the whole \(E(a,b;r)\), then \(f\) is a restriction of a solution of an equation of type \(f(x+y)=Kf(x)f(y)\) with some \(K\neq 0\). A related extension theorem is proved. 2. If \(f(x_0,y_0)=0\) for some \((x_0,y_0)\in E(a,b;r)\), then 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.