Why are curve reparametrization functions required to be monotone?

Question

Suppose one has a parametrized smooth plane curve $x = f(t)$, $y = g(t)$. One may reparametrize it, let's say, by making a substitution $t = z(u)$. Standard elementary differential geometry books then remark that $z$ should be monotone. Why do they require this? For example, if we had a curve with parameter $t$, with $t$ going from $[0,2)$ what would be the problem of setting $t = u^2$, with $u$ going from $(-\sqrt2,\sqrt2)$? Would it not produce the same image?

Answer 1

Among some of the complexities which a non-$``1$-$1"$ function might introduce, you might consider that if the same value of $t$ is assigned to different values of $u$, this might result in doubly-assigning values of the derivative (w.r.t. to the new parameter $u$) at the same point of the curve. So, from the analytic-viewpoint, this might introduce singularities and incosistencies on the definition of the tangent space, even at points where the curve is -geometrically- smooth.