Let \(G \subset\mathbb C\) be a domain and \(\zeta\in \partial G\). A function \(\varphi\) is called \(T_\zeta\)-universal on \(G\) if it is holomorphic on \(G\) and satisfies the following property: For all compact sets \(K\) with connected complement and all complex valued functions \(f\) on \(K\) there exist sequences \(\{a_n\}\) and \(\{b_n\}\) with \(a_nz + b_n\in G\) \(\forall z\in K\) and \(\forall n \in\mathbb N,\) \(a_nz + b_n\to\zeta\) \(\forall z\in K\), \(n\to\infty\), \(\varphi(a_nz + b_n)\to f(z)\) uniformly on \(K\), \(n\to\infty\). It is a natural question whether a holomorphic function \(\varphi\) can exhibit universal properties in the sense that it is \(T_\zeta\)-universal on \(G\) for all \(\zeta\) of a prescribed subset \(E\subset\partial G\) and such that \(\varphi\) is not \(T_\zeta\)-universal for all \(\zeta\) on the complementary boundary part \(\partial G \setminus E\). If a closed set \( E\subset \partial \mathbb D\) is prescribed then it is proved that there exists a function \(\varphi\) which is \(T_\zeta\)-universal for all \(\zeta\in E\) and is not \(T_\zeta\)-universal for all \(\zeta\in F :=\partial \mathbb D\setminus E\). In addition the function \(\varphi\) can be chosen such that \(\varphi\in C^\infty(F)\) and has a lacunary power series expansion \(\varphi(z) =\sum_{n=0}^\infty a_nz^n\quad\text{with}\quad a_n = 0\quad\text{if}\quad n\notin Q,\) where \(Q\) is a subsequence of \(\mathbb N_0\) with upper density \(\overline d(Q) = 1\). The construction of such a function follows by an inductive process, where a lemma on lacunary polynomial approximation and the existence of special \(T_\zeta\)-universal functions on starlike domains are the essential tools.