Discrete vs Continuous?

Question

Mathematics

The question is, is there a discrete two-dimensional function, what is there to hide, an image, there are a lot of theories for the analysis of two-dimensional continuous functions, how to make the transition from beautiful continuous theory to discrete practice? Particularly interested in such things as partial derivatives of the first and second orders. Which way to look / read?

Answer 1

Continuous and everywhere defined functions are needed only to take an infinitesimal sampling step. And here this step is quite large and clearly known. In general, it is not clear to me which formulas do not fit.

Answer 2

how to make the transition from beautiful continuous theory to discrete practice? Particularly interested in such things as partial derivatives of the first and second orders.

1. Interpolate and use as continuous.
2. Stay within the discrete. For example, derivatives are considered somewhat more convenient. But they will also be discrete, of course, so it depends on what, of course.

how to take the derivative with respect to dxdy if there is a two-dimensional matrix?

If it is necessary at each point on a continuous set, then interpolate. If it is necessary only at the starting points, then it is clear that it is possible to work discretely.
The task, in short, is somehow incomprehensibly set.