f(x+y) = f(x)*f(y) for all real x,y. Setting x = y = 0: f(0) = f(0)^2, so f(0) = 0 or 1. If f(0) = 0, then f(x) = 0 for all x (trivial). For non-trivial: f(0) = 1, and f(x) = $a^x$ where a = f(1). Given f(1) = k, then f(n) = $k^n$ for integer n, extending to f(x) = $k^x$. Note: f(x) > 0 always (since f(x) = f$\frac{x}{2}$^2 >= 0).