 Whole-plane reduction for continuous superposition

For $f\in C({\mathbb{R}}^2)$, choose a positive continuous function $A$ with $A(t)\ge (1+t){\max }_{x^2+y^2\le t}|f(x,y)|$ for $t\ge 0$. Such an $A$ is obtained by piecewise linear interpolation of a sufficiently large increasing sequence. Then $b(x,y)=f(x,y)/A(x^2+y^2)$ tends uniformly to zero at infinity. Transport $b$ to $(0,1{)}^2$ by $x=\tan (\pi (s-1/2))$ and extend it by zero on the boundary; the extension is continuous. Apply the compact Kolmogorov-Arnold representation theorem, then multiply by $A(x^2+y^2)$. Multiplication introduces no additional primitive operation because $uv=[(u+v{)}^2-(u-v{)}^2]/4$ uses only addition and continuous one-variable squaring and scaling.