I am reading a book that somehow gives two different definitions for a smooth curve, so I wonder which one is more precise.
Suppose we're given the curve $L$ defined by the parametric equation $$(x,y)=(f(t),g(t)),\quad t\in[a,b].$$
The first definition I encountered is that this curve is said to be smooth if $f'$ and $g'$ exist in the interval $[a,b]$ and also they are not simultaneously zero at any point $\theta \in [a, b]$.
The second definition I encountered is that this curve is said to be smooth if $f'$ and $g'$ exist in the interval $[a,b]$ and they are also continuous in $[a,b]$. I think the definition given here also matches my second definition.
https://mathworld.wolfram.com/SmoothCurve.html
But to me the first definition makes more sense because smooth somehow seems to be mean that the curve has a tangent line at any point. Isn't that what smooth is?
So I am very confused here.
Do we need to require $f'$ and $g'$ to be continuous?
Do we need to require that $f'$, $g'$ are both simultaneously zero at any point $\theta \in [a,b]$?
Also, what happens really if $f'$ and $g'$ are simultaneously zero at some point? I mean, the book says that then the curve has no tangent at that point. OK, but how does it look like at that point? Does it have more than one tangent there, or no tangent at all? Any examples?
I also constructed two examples myself. Here they are.
Is this curve smooth? Why? $$(x,y) = (t^5 \cos(t), t^2),\quad t \in [-\pi, \pi]$$
What about this one? Why? $$(x,y) = t^2 \cos(t), t^5),\quad t \in [-\pi, \pi]$$
For $t=0$ both $f'$ and $g'$ are equal to zeros (for both curves).
Please someone more knowledgeable clear these confusions.
Note that I am asking just in a normal calculus (real analysis) context. I mean, I don't need complicated generalizations, just an explanation in this particular context.