I am wondering wether it exists a theorem that any continuous path on the plane one can
approximate with algebraic curve $P(x,y)=0$ ($P$- is a polynom)?



Algebraic curve approximation
Given any compact set $K$ in the plane (in particular the image of a compact interval under a continuous function) and $\epsilon > 0$, there is a finite set $\{(x\_j, y\_j)\}\_{j=1}^n \subseteq K$ such that $K$ is contained in the union of the disks of radius $\epsilon$ centred at $(x\_j, y\_j)$. Then $K$ is within distance $\epsilon$ in the Hausdorff metric
of the real algebraic curve $P(x,y) = 0$, where $P(x,y) = \prod\_{j=1}^n ((x-x\_j)^2 + (y-y\_j)^2 - \epsilon^2)$. 


