How can I prove that Game of Life's evolution function is continuous?

Let $N = \{-1,0,1\}^2$ denote the neighborhood of the cell at the origin and, for any set $S \subset \mathbb Z^2$ of cell coordinates, let the Minkowski sum $$S + N = \{(i+a,j+b) : (i,j) \in S, (a, b) \in N \}$$ denote the combined neighborhood of all the cells in $S$. Then $$x_{S+N} = y_{S+N} \implies f(x)_{S} = f(y)_{S}.$$

Since, in particular, $[-k,k]^2 + N = [-k-1,k+1]^2$, it follows that $$x_{[-k-1,k+1]^2} = y_{[-k-1,k+1]^2} \implies f(x)_{[-k,k]^2} = f(y)_{[-k,k]^2},$$ and thus that $$d(x,y) \le 2^{-k-1} \implies d(f(x),f(y)) \le 2^{-k}.$$ Therefore, for any $\epsilon > 0$, $$d(x,y) < \frac\epsilon2 \implies d(f(x),f(y)) < \epsilon.$$

Note that this proof does not depend on the specific evolution rule of Conway's Game of Life in any way, except for the fact that the evolution of each cell is solely determined by the states of the cells in its neighborhood. The same proof applies just as well to any other deterministic cellular automaton defined on the 3×3 cell Moore neighborhood, and can be straighforwardly generalized to any deterministic cellular automaton defined on any finite neighborhood.