The author proves the existence of extensions of a given symmetric Feller process \(Z_t\), from \(\mathbb{R}\backslash\{0\}\) to \(\mathbb{R}\), depending on a parameter \(\alpha\in[0,1]\): if \(\alpha=1/2\), the extension exists always; if \(\alpha\neq 1/2\), the extension exists if and only if \(Z_t\) does not jump over \(0\).