Definition 5.7 has $a^n = 1$ if $n=0$ and $a \neq 0$. Please consider dropping the $a \neq 0$ condition --- it would be better to have $a^n = 1$ if $n=0$, for all real $a$.
True, the function $F(a,n) = a^n$ is discontinuous at the point $(a,n) = (0,0)$; the famous "indeterminate form" $0^0$. But the value at that exact point should be $0^0 = 1$.
