>Someone knows how could I proof that only the exponential
>function follows that f(a+b)=f(a)*f(b)?
This is only true if the function f is assumed to be continuous and defined for all reals. Please note that, if g(x) = log(f(x)) (and the definition makes sense, i.e., if f(x) is positive) then
g(a + b) = g(a) + g(b).
For the latter see https://www.cut-the-knot.org/do_you_know/linear.shtml