Differentiable curve in Arnold's book

Question

In the book "Mathematical Methods of Classical Mechanics" of Arnold at pag.7 it's said that the trajectory of a differentiable motion (a motion is defined previously as a curve $ x:I\subset\mathbb{R}\rightarrow\mathbb{R^n} $) on a plane can have the shape drawn in the figure below. In my opinion the red arrows (drawn by me: not belonging to original figure) show that in that point the motion is not differentiable because the right and the left limit are different in that point. The definition adopted by Arnold for differentiability is the classical one for curves. Where am I wrong?

Answer 1

Yes it is possible. Keep in mind you are differentiating with respect to time $t$, and that the derivative of position is velocity. If the point slows down coming to that sharp point $p$, has a velocity of magnitude $0$ hitting the point $p$, and then accelerates in a new direction after leaving $p$, then the right limit [just after leaving $p$] and the left limit [just before right up to arriving at $p$] of the velocity is the same--$0$.

Automobiles make $3$-point turns, while being capable of only motion that is differentiable.