## Cellular Automata and Continuous Functions: Negative Results

**Yuri Ozhigov***Department of Mathematics,**Moscow State Technological University "Stankin,"**Vadkovsky per. 3a, 101472, Moscow, Russia*

#### Abstract

Let be a finite alphabet, , and .

A configuration is a function of the form: , and is the set of all configurations. Two configurations and are near if is small, where .

The following results are proved.

- There is no sequence of functions such that and uniformly converge to continuous functions in such a topology.
- Evolutions of cellular automata (CA) cannot be approximated by the superpositions of real continuous functions.

In the proofs of these results advantage was taken of some CA acting in and in with a stationary boundary condition.