# Is the role of the boxed condition $z'(t)\neq 0$ to avoid going back?

The role of the boxed condition $$z'(t)\neq 0$$ is to avoid going back, isn't it?

No. It is so that the velocity is never $$0$$, which implies that we can parametrize the curve by the arc length. It also implies that $$z\bigl([a,b]\bigr)$$ has no “corners”, which corresponds to the idea of a smooth curve.