  The existence and the uniqueness of the solution to the BCS gap equation of
superconductivity is established in previous papers, but the temperature
dependence of the solution is not discussed. In this paper, in order to show
how the solution varies with the temperature, we first give another proof of
the existence and the uniqueness of the solution and point out that the unique
solution belongs to a certain set. Here this set depends on the temperature
$T$. We define another certain subset of a Banach space consisting of
continuous functions of both $T$ and $x$. Here, $x$ stands for the kinetic
energy of an electron minus the chemical potential. Let the solution be
approximated by an element of the subset of the Banach space above. We second
show, under this approximation, that the transition to a superconducting state
is a second-order phase transition.
