One of long-standing problems in mathematical studies of superconductivity is
to show that the solution to the BCS gap equation is continuous in the
temperature. In this paper we address this problem. We regard the BCS gap
equation as a nonlinear integral equation on a Banach space consisting of
continuous functions of both $T$ and $x$. Here, $T (\geq 0)$ stands for the
temperature and $x$ the kinetic energy of an electron minus the chemical
potential. We show that the unique solution to the BCS gap equation obtained in
our recent paper is continuous with respect to both $T$ and $x$ when $T$ is
small enough. The proof is carried out based on the Banach fixed-point theorem.