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The first aim of this work is to establish a Peano type existence theorem foran initial value problem involving complex fractional derivative and the secondis, as a consequence of this theorem, to give a partial answer to the localexistence of the continuous solution for the following problem withRiemann-Liouville fractional derivative: \begin{equation*} \begin{cases}&D^{q}u(x) = f\big(x,u(x)\big), \\ &u(0)=b, \ \ \ (b\neq 0). \\ \end{cases}\end{equation*} Moreover, in the special cases of considered problem, weinvestigate some geometric properties of the solutions.

Complex variable approach to analysis of a fractional differential equation in the real line