  We obtain a strong renewal theorem with infinite mean beyond regular
variation, when the underlying distribution belongs to the domain of geometric
partial attraction a semistable law with index $\alpha\in (1/2,1]$. In the
process we obtain local limit theorems for both finite and infinite mean, that
is for the whole range $\alpha\in (0,2)$. We also derive the asymptotics of the
renewal function for $\alpha\in (0,1]$.
