Consider homogeneous polynomials $P(X,Y)$ in two variables. According to Approximation by homogeneous polynomials by Totik, "every even continuous function on a centrally symmetric convex curve can be uniformly approximated by homogeneous polynomials".
My question is the following: can all (even?) continuous functions be approximated by homogeneous rational functions with numerator and denominator of the same degree? If not, which can be?