Here, an appropriate extension of the so-called one-sided Lipschitz condition is presented in case of discontinuous functional-differential inclusions \[ \dot x(t)\in F(t,x_t), \,\, x_0=\phi, \,\, t\in [0,1]. \] The new condition is called strong one-sided Lipschitz. The most important properties of the one-sided Lipschitz ordinary differential inclusions are extended to the case of strong one-sided Lipschitz functional-differential inclusions. Moreover, an exponential formula for the reachable set is obtained.