Some analysis on a fractional differential equation involving a noncontinuous right-hand side

By developing new techniques we establish local existence and uniqueness theorems for an initial value problem involving a nonlinear equation in the sense of Riemann-Liouville fractional derivative in the case that the nonlinear function on the right hand side of the equation is not continuous on $[0,T]\times \mathbb{R}\mathrm{.}$