The unique solvability issues of the Cauchy problem with a fractional derivative are considered in the case when the coefficient at the desired function is a continuous function. The solution of the problem is found in an explicit form. The uniqueness theorem is proved. The existence theorem for a solution to the problem is proved by reducing it to a Volterra equation of the second kind with a singularity in the kernel, and the necessary and sufficient conditions for the existence of a solution to the problem are obtained.