On /()X\)-convex functions

Let $X$ be a Banach space. Let $f(\cdot )$ be a real valued function defined on an open convex set $\mathrm{\Omega }\subset X^{\ast }$, where $X^{\ast }$ as usual denote the conjugate space. We say that the function $f(\cdot )$ is $X\$$convex, if there is a set ${\mathrm{\Phi }}_f\subset X$ such that $$f(x^{\ast })=su{p}_{x\in {\mathrm{\Phi }}_f,r\in \text{R}}{x}^{\ast }(x)+r.\text{eqno}(1)$$ In the paper it will be shown that if $X$ is separable, then the function $f(\cdot )$ is Frechet differentiable on a dense $G_{\delta }$ set.