I encountered this theorem some time ago.
I think I was able to prove it pretty easily.
But it made me thinking about the following.
For some curve and for its parametrization, could we have that at some fixed point $s = s_0$ the curvature $\kappa(s_0) = 0$ but there is no neighborhood $U$ around $s_0$ in which $\kappa(s)= 0$ in the whole neighborhood $U$?!
What is the answer to this question?
And what is the justification of it?