subdifferentiable mapping

Let $X$ be a Banach space, and let $X^{*}$ be the dual space of $X$.For a function $f:X\to ℝ$, and $x\in X$, letus define

$$\partial f(x)=\{r^{*}\in X^{*}:f(x)-f(y)\le r^{\ast }(x-y)\text{for all}y\in X\}.$$

If $\partial f(x)$ is non-empty, then $f$ is subdifferentiableat $x\in X$, and if $\partial f(x)$ is non-empty for all $x$, then$f$ is subdifferentiable [1, 2].