F-injectivity and Buchsbaum singularities

Let (R,m) be a local ring that contains a field. We show that, when R has equal characteristic p>0 and when H_m^i(R) has finite length for all i<dimR, then R is F-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we show that such an R is in fact a Buchsbaum ring. This answers positively a question of S. Takagi that F-injective singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. We also study the characteristic 0 analogue of this question and we show that Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum in the graded case.