Suppose $X$ is a Banach space and $f$ is a Lipschitz Gateaux differentiable function on $X$. Let $S : [0,1]^{\mathbb{N} \backslash \{ n \}} \rightarrow X$. Then we have
$$f(x_n + S(t)) - f(S(t)) = \int_0^1{\langle \nabla f(ux_n - S(t)), x_n \rangle} du$$
The equation above is taken from here, page $128$.
Question: what is the definition of $\nabla f(ux_n - S(t))$ and why requires $f$ to be Lipschitz Gateaux differentiable?
I need to understand it so that I may know how the author obtains $\| \nabla f(y) \|_{X^*} \leq \| f \|_{Lip}$.