This paper presents an online self-regulating clustering algorithm (SRCA) to construct parsimonious radial basis function networks (RBFN) for function approximation applications. Growing, merging and splitting mechanisms with online operation capability are integrated into the proposed SRCA. These mechanisms enable the SRCA to identify a suitable cluster configuration without a priori knowledge regarding the approximation problems. In addition, a novel idea for cluster boundary estimation has been proposed to effectively maintain the resultant clusters with compact hyper-elliptic-shaped boundaries. Computer simulations show that RBFN constructed by the SRCA can approximate functions with a high accuracy and fast learning convergence. Benchmark examples and comparisons with some; existing approaches have been conducted to validate the effectiveness and feasibility of the SRCA for function approximation problems.