In Gelfand Calculus of Variation, chapter 1.7, the variational derivative is defined as:
$\frac{\partial J}{\partial y}|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\Delta\sigma}$, where $h(x)$ is different from zero only in the neighborhood of $x_0$. Also $\Delta\sigma$ is the area lying between the curve $h(x)$ and the x-axis, and $\Delta\sigma$ goes to zero in such a way that both $\max|h(x)|$ and the length of the interval in which $h(x)$ is non-vanishing goes to zero
I wonder if there exist a more precise statement for this definition and if there is any characterization of when such a derivative exist. Also, is there any relation between the variational derivative and the variation (or differential) of a functional defined in Gelfand chapter 1.3.2.
Moreover, what is the purpose of this definition?