# Is there a discontinuous surface in 4d space with a constant 2d derivative that can be attached to a 2d grid in that space?

Given a 3d space with a 2d grid in it, one can imagine every grid cell to contain a wedge shape, such that when sampling across the wedge shape, you get a constant nonzero derivative. Imagine the two opposite bottom edges perpendicular to that axis of the wedge shape (parallel along the wedge) to attach to the grid.

Given a 4d space, is it possible to create a surface that attaches to all 4 edges of a 2d grid cell, and have a constant nonzero derivative in both directions of the grid, like the wedge example in 3d space has in one direction of the grid?

I am writing a program to 2d sample 4d continuous noise, aiming to be able to increase and decrease the detail locally, zooming without losing continuity so to speak. Sampling along a wedge shape increases the distance sampled (decreasing detail) while ending at the same z coordinates at every grid cell boundary, preserving continuity. I hope a 4d shape allows the same for two axes?

The main use of the grid is to prevent looping of the noise and ease of implementation. One can sample along various shapes in the space directly instead of affixing a 2d grid, but those might intersect given multiple or loop around. Approaches that do not involve grids and are not infinite in 2d are also welcome, although it might make it harder to implement.

Apologies for not having the proper mathematical knowledge to describe the problem. I hope your answers will help me find the right vocabulary as much as a solution.