Function subgroups \(G_ 1\), \(G_ 2\) of any Kleinian group \(G\) are investigated, and some results in \textit{B. Maskit} [Ann. Math. Studies No. 79, 349-367 (1974; Zbl 0305.30021)] are generalized. Theorem 1 shows that any function group is geometrically tame by using the results in \textit{F. Bonahon} [Ann. Math., II. Ser. 124, 71-158 (1986; Zbl 0671.57008)]. This theorem implies that \(G_ 1\cap G_ 2\) is finitely generated (Theorem 2). let \(\gamma\in G\) be an element with \(\hbox{fix}(\gamma)\cap\Lambda(G_ 1)\neq\emptyset\), where \(\Lambda(G_ 1)\) is the limit set for \(G_ 1\). If \(\gamma\) is loxodromic, then \(\gamma^ n\) is contained in \(G_ 1\) for some \(n\in\mathbb{N}\), and if \(\gamma\) is parabolic, then there exists a \(\lambda\in G_ 1\) with \(\hbox{fix}(\gamma)=\hbox{fix}(\lambda)\) (Theorem 3). In Theorem 4, the necessary and sufficient condition for the equality \(\Lambda(G_ 1\cap G_ 2)=\Lambda(G_ 1)\cap\Lambda(G_ 2)\) is given. In particular, this equality holds if \(G_ 1\) contains no parabolic elements.