quantum mechanics - Does Feynman path integral include discontinuous trajectories?

While reading this derivation of relation of Schrödinger equation to Feynman path integral, I noticed that $q_i$ can differ form $q_{i+1}$ very much, and when the limit of $N\to\infty$ is taken, there remain lots of paths, which are discontinuous (almost) everywhere — i.e. paths consisting of disconnected points.

Do I understand this wrongly? How do such discontinuous paths disappear on taking the limit? Or maybe they have zero contribution to integral?