The algorithm for identifying whether or not a parametric curve is smooth contains a first step of finding out when dx/dt and dy/dt is simultaneously 0. What are the other steps?
My textbook points at that there are functions where dx/dt and dy/dt are both 0 and still smooth. It never gives a rigorous explanation.
In addition, when looking at the solutions to the problem set of the same textbook, the author when proving that one of the functions is not smooth looks at whether or not the sign of dx/dt or dy/dt shifts whenever they are both 0.
However, this way of proving smoothness does not transfer to linear functions with a sharp upward bend at a single point. dx/dt and dy/dt never changes signs in this case.
What is a rigorous definition of smoothness for parametric functions? How do you prove whether or not a function is smooth?
