Wave Equation

This is just a bit over spill as we were covering the wave equation in school. There is an interesting point although. If we look at wave equations with a time derivative and a space derivative . Now we have the following curious fact: if we assume that the solution of the wave equation should preserve energy, should satisfy a locality condition and have both space and time to be locally compact then is discrete if and only is discrete. The reason is that if is discrete then the solution of the wave equation is entire in t which means that if two points are separated then we have have the solution constant in t for some time meaning that the solution would be constant zero. I might cover how to realize the discrete to have all the properties: locality and energy conservation.