The purpose of this paper is to give a description of the space \(A_ D(F)\) of functions on a relatively closed subset F of an open plane set D which can be approximated uniformly on F by functions in H(D), i.e. functions analytic on D. The main result is the decomposition \[ A_ D(F)=C_{ua}(F\cup \Omega (F))+H(D)). \] Here \(\Omega (F)=D\setminus (F\cup M_ F)\), where \(M_ F\) is the set of all \(z\in D\setminus F\) which can be joined to \(D^*\setminus D\) by an arc in \(D^*\setminus F\) \((D^*\) is the one point compactification of D) and \(C_{ua}(F\cup \Omega (F))\) denotes the uniformly continuous functions on \(F\cup \Omega (F)\) which are analytic in the interior of \(F\cup \Omega (F).\) Basic ingredients in the proof are Vitushkin's scheme for rational approximation [see e.g. \textit{T. W. Gamelin}, Uniform Algebras (1969; Zbl 0213.40401)], together with Arakelyan's noncompact versions of Mergelyan's classical polynomial approximation theorem [see e.g. \textit{D. Gaier}: Vorlesungen über Approximationen im Komplexen (1980; Zbl 0442.30038)] and earlier related results by the author.