  This paper is devoted to a nonlinear singular Riemann-Liouville type
fractional differential equation, the local existence of whose continuous
solutions under the weakest condition remained as an open problem until now.
The singularity of the equation arises from the discontinuity of the right-hand
side function $f(x,\omega)$ at $x=0$, and its order of singularity is the same
as the order of the fractional differential operator in the equation. The local
existence of solutions to singular equations of such type cannot be established
by a direct application of fixed point theorems only or other methods even
though they are enough for the same purpose in the case of equations including
a singularity of the order less than the order of the corresponding fractional
R-L operator. For this reason, we here propose a novel technique with a result
for continuous functions to establish a local existence theorem for the
problem. Moreover, a continuation result is suggested for the local solutions
to the problem and, thus, it will be revealed that in what circumstances their
existing interval can be extended to larger ones. Thanks to this and the
proposed technique, the global existence theorem for both sublinear and linear
equations is achieved.
