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In the preceding work \cite{watanabe3}, it is shown that the solution to theBCS gap equation for superconductivity is continuous with respect to both thetemperature and the energy under the restriction that the temperature is verysmall. Without this restriction, we show in this paper that the solution iscontinuous with respect to both the temperature and the energy, and that thesolution is Lipschitz continuous and monotonically decreasing with respect tothe temperature.

Lipschitz continuity and monotone decreasingness of the solution to the BCS gap equation for superconductivity