Why do we need non-zero derivative to define a regular curve?

Question

In my calculus course, I've been taught that to define a regular curve on an interval we state that it is continuous and differentiable with non-zero first derivative. While I can see why we need continuity and differentiability, I fail to see why we'd need the derivative to be different than zero.

Answer 1

Because it is convenient that a curve is, after a suitable reparametrisation, be a curve such that the scalar velocity (that is, the norm of the velocity) is $1$, and being regular ensures that that occurs.

Besides, being regular prevents us from findind curves with corners, such has$$\begin{array}{rccc}c\colon&[-1,1]&\longrightarrow&\mathbb{R}^2\\&t&\mapsto&\begin{cases}(t^2,t^2)&\text{ if }t\geqslant0\\(-t^2,t^2)&\text{ otherwise,}\end{cases}\end{array}$$which is shapped like a letter V.

Answer 2

Imagine that you move fast at first, then gradually slow down, finally stop moving, and move again in another direction.

Your position is continuous with respect to time, and your velocity exists all the time. However, your trajectory has a corner where you stopped moving.

Therefore if a curve has zero derivative somewhere, you can't be sure if it is regular (in the sense that it has no corners).