I've been studying vector calculus, and I came across the notion of regular smoothness.
In particular, the textbook I am using claims that $y = |x|$ has no regular smooth parametrization. No proof of this fact is given.
This is intuitively obvious to me; here is why:
Given any parametrization of this curve, the derivative at $(0,0)$ must be the zero vector. For suppose we are given $$ r(t) = \langle x(t), |x(t)| \rangle $$
Then we get that $r'(t)$ does not exist if $x'(t) \neq 0$, because then the directional limits would disagree on sign. Thus the only way for $x'(t)$ to be defined is to be the zero vector. Thus this curve has no regular parametrization.
Would this constitute a proof in this scenario? -- Furthermore, how can I generalize this to determine which classes of curves are regular smooth. For example, are 'cusps' the only issue?