In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s+ is better than that at 0+, where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L1 numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [2/3, 1), α is the order of fractional derivative. Furthermore, an improved fitted L1 method is proposed and the region of optimal convergence order is larger. For the case t > s, stability and min { 2α, 1} order convergence of the fitted L1 scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.