Dependence of differential properties of a function on the speed of its rational approximations in the metrics of $L_p$

We establish the existence almost everywhere on $[0,1]$ of a generalized differential of a given order for a function which can be sufficiently well approximated by rational functions in the metrics of $L_p[0,1]$ ($0<p\le\infty$); we will clearly express the metric dimension of the set of points at which the function is not differentiable.