This paper investigates the L₂-gain analysis and synthesis issues under the Filippov framework for a class of nonlinear time-delay systems with discontinuity. First, an extension of L₂-gain property along all the Filippov solutions is given for the time-delay systems, and a sufficient condition is presented that guarantees the nonlinear time-delay systems is strongly stable in the sense of Filippov solution, and the L₂-gain less than a given level. The condition is described by functional partial differential inequality. Then, as an application of the presented condition, a feedback control design problem is solved that the controller strongly stabilized a given plant with pre-specified L₂-gain. Furthermore, as a special case study, the case of linear systems is discussed and it is shown that for the piecewise continuous linear systems, the sufficient condition is represented by algebraic Riccati inequalities.