  Information decompositions quantify how the Shannon information about a given
random variable is distributed among several other random variables. Various
requirements have been proposed that such a decomposition should satisfy,
leading to different candidate solutions. Curiously, however, only two of the
original requirements that determined the Shannon information have been
considered, namely monotonicity and normalization. Two other important
properties, continuity and additivity, have not been considered. In this
contribution, we focus on the mutual information of two finite variables $Y,Z$
about a third finite variable $S$ and check which of the decompositions satisfy
these two properties. While most of them satisfy continuity, only one of them
is both continuous and additive.
