Conditions for regularity of increases and densities of the renewal function for distributions without first moment

The asymptotic of renewal functions generated by distribution $F(t)$ with regularly varying tails $F(-t)+1-F(t)$ of order $\beta\in(0,1]$ without the first moment may be calculated by Karamata's Tauberian theorem. The asymptotic behavior of increments for such type of renewal function in non-arithmetical case was described by Erickson (1970, 1971). The situation is the following: for $\beta\in(0.5,1]$ the increments are asymptotically equivalent to the increment of argument multiplied by formal derivative of renewal function (without derivative of slowly varying function). For $\beta\in(0,0.5]$ it does not hold true. V. Vatutin and the author find the condition $F(-t+\Delta)-F(t+\Delta)-F(-t)+F(t)=O(\Delta)(F(-t)+1-F(t))/t, t\to\infty,$ sufficient for formal differentiation of the renewal function also in the case $\beta\in(0,0.5]$.
For absolutely continuous distributions we find sufficient conditions for the asymptotic behavior of the first and second derivatives of the renewal function to correspond to formal procedures.