  Based on the proofs of the continuity of the conditional entropy by Alicki,
Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF)
method. This method allows us to prove a great variety of continuity bounds for
the derived entropic quantities. First, we apply the ALAFF method to the
Umegaki relative entropy. This way, we recover known almost tight bounds, but
also some new continuity bounds for the relative entropy. Subsequently, we
apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This
yields novel explicit bounds in particular for the BS-conditional entropy, the
BS-mutual and BS-conditional mutual information. On the way, we prove almost
concavity for the Umegaki relative entropy and the BS-entropy, which might be
of independent interest. We conclude by showing some applications of these
continuity bounds in various contexts within quantum information theory.
