The first one is right. A certain point of a binary function has a directional derivative in any direction, so it cannot be inferred that the function is continuous at that point.

The first paper is correct. Let y = kx (1/3), we can know that the limit is related to k when x approaches 0, and the limit of the function at zero does not exist, so it is discontinuous.

There is something wrong with the penultimate paragraph of the second paper. This function is continuous along any straight line, so the following definition of continuity cannot be derived. This step is groundless.