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
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.
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