  The main objective of this article is to discuss the local existence of the
solution to an initial value problem involving a non-linear differential
equation in the sense of Riemann-Liouville fractional derivative of order
$\sigma\in(1,2),$ when the nonlinear term has a discontinuity at zero.
Hereafter, by using some tools of Lebesgue spaces such as H\"older inequality,
we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness
theorems for the problem.
