All statements of the four-vertex theorem I've seen so far talk about simple, closed, smooth plane curves. And the theorem itself is about the (extrema of the) curvature of these curves. What I haven't seen yet is a justification why we can assume that the curve's curvature is everywhere defined. After all, there are smooth plane curves which don't admit a regular parametrization, right?
Does the theorem only hold for curves with a regular parametrization? Or does the combination "smooth + closed + simple + plane" somehow imply that we can always find a regular parametrization?
The answer is that the four vertex theorem only holds for curves with a regular parametrization. This is one common definition of "smooth curve" in mathematics, though it isn't the only one.