The authors consider the following situation: Let \(f:[a,b] \to\mathbb{R}\) be a continuous but not a rational function. By \(r_n\) the best uniform rational approximation of \(f\) of degree \((n,n)\) is denoted. The paper is concerned with the convergence of the sequence \(\{r_n\}_{n\in\mathbb{N}}\) (or, more precisely, of the continuations of these functions) in the complex plane. The main result says that \(\{r_n\}_{n\in\mathbb{N}}\) converges in a weak sense uniformly on compact sets in \(\mathbb{C}\), provided that certain specific conditions, too technical to be given here, are satisfied. Moreover, several consequences of this result are formulated.