The first part of the present paper is devoted to an alternative proof of the existence of a unique solution to the BCS gap equation. Next, it is defined a certain subspace
of a Banach space consisting of continuous functions and it is considered the solution approximated by an element of
. Next, it is shown that, under this approximation, the transition to a superconducting state is a second-order phase transition. In other words, it is established that the condition that the solution belongs to
is a sufficient condition for the second-order phase transition to superconductivity.