Let A be a bounded domain whose boundary S is a closed Jordan curve. If $\Sigma \in C^{1,h}$, a necessary and sufficient condition has been given for the uniform approximation of a function $f$ holomorphic in $A$ and continuous in $A \cup \Sigma$, by rational functions with prescribed poles. In this paper one proves that the theorem fails to be true under the only assumption $\Sigma \in C^{1}$.