On extension of the Legendre transform on $C(X)$ and its applications

In some applications there are functionals on $C(X)$, which is the the space of continuous functions on a compact topological space $X$, for which variational principles apply. These variational principles represent the functional $f$ as a Legendre transform of some functional $g$ on the dual space. In applications the function $a(x)=\exp\varphi(x)$ usually has physical interpretations and the functional $f$ generates the functional $f(\ln a)$. The latter is defined on the cone of positive continuous functions. In the paper the questions of the extension of this functional and the variational principle taking place for such an extension are raised. The main difficulty is that the functional $f(\ln a)$ can have no continuous prolongation. The proof is based on the generalization of the classical Dini theorem, which was obtained in the process.