This article presents a method for the explicit construction of limit state functions using Support Vector Machines (SVM). An algorithm is proposed for updating the SVM decision function by carefully selecting the training samples. This results in the construction of an accurate limit state function with a reduced number of function evaluations. Specifically, the SVM-based approach aims at handling the difficulties associated with the reliability assessment of problems exhibiting discontinuous responses and disjoint failure domains. The explicit construction of limit state functions allows for an easy calculation of a probability of failure and enables the association of a specific system behavior with a region of the design space. Three problems are presented to demonstrate the explicit construction of a limit state function. The proposed update scheme is validated by comparing the obtained explicit function to actual analytical limit state functions.