I know that the Weierstrass approximation theorem says the following: Let $f:[0,1]\to \mathbb R$ a continuous function then for all $\varepsilon>0$ exists a polinomial function $p$ such that $\|f-p\|<\varepsilon$.
I was wondering if I have a non-empty compact convex subset $D$ in $\mathbb R^2$ and a continuous mapping $X:D\to D$ such that $X(x,y)=(f(x ,y),g(x,y))$. Then it is possible that there exist polynomials $p,q\in \mathbb R[x,y]$ such that $\|(f(x,y),g(x,y))-(p(x,y), q(x,y))\|<\varepsilon$ for all $(x,y)\in D$?
Could someone tell me if this version would be valid? Perhaps a bibliographical reference or a proof would be very helpful for me. Thanks in advance.