A parametrized curve is a function $z(t)$ which maps a closed interval $[a, b] \subset \mathbb{R}$ to the complex plane. We shall impose regularity conditions on the parametrization which are always verified in the situations that occur in this book. We say that the parametrized curve is smooth if $z'(t)$ exists and is continuous on $[a, b]$, and $z'(t) \neq 0$ for $t \in[a, b]$.
Above is the definition of smooth parameterized curves given in Stein and Shakarchi's Complex Analysis.
Questions:
(i) Why do we require $z'(t) \not= 0$? I have not seen this condition used in any proofs.
(ii) Where is the "regularity condition" imposed - I thought this meant $z(t) \in C^{\infty}[a,b]$ (?)