We had the following theorem and I have two questions about it:
Definition A curve $c:I \to \mathbb{R}^3$ is said strictly regular, if $c'(t)$ and $c''(t)$ are linearly independent.
Theorem Let $c$ be a strictly regular curve. Then $$T=\frac{c'(t)}{\|c'(t)\|}\qquad B=\frac{c'\times c''}{\|c'\times c''\|}\qquad N=B\times T$$ $$\kappa(t)=\frac{\|c'\times c''\|}{\|c'\|^3}\qquad\tau(t)=\frac{\langle c'\times c'',c'''\rangle }{\|c'\times c''\|^2}$$
Is it true that if $c$ is strictly regular $\implies$ $c$ is a regular curve? My argument would be because if $c'(t)=0$ then $c''(t)$ is always linearly dependent to the zero-vector.
Does the theorem also hold for just regular curves or even just curves? If not, which formulas do still hold?
The answer to your first question is yes, of course. Any set of vectors with the $0$-vector among them is linearly dependent.
Assuming you have a regular curve, there will be no principal normal (hence no $N$ or $B$) defined at points where $\kappa=0$. These are precisely the points where $c',c''$ become linearly dependent. So the whole game comes to a grinding halt. You still are fine with the formula for $T$ and we do get $\kappa=0$. But then we can go no farther.