In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem D_t ^αu(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x₀ on a Banach space X with order α ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if A generates an analytic α-times resolvent family on X and f ∈ L^p ([0,T];X) for some p > 1/α, then the mild solution to the above equation is in C^α-1/p[ò,T] for every ò > 0. Moreover, if f is Hölder continuous, then so are the D_t ^αu(t) and Au(t).