  This paper addresses the correspondence between linear inequalities of
Shannon entropy and differential entropy for sums of independent group-valued
random variables. We show that any balanced (with the sum of coefficients being
zero) linear inequality of Shannon entropy holds if and only if its
differential entropy counterpart also holds; moreover, any linear inequality
for differential entropy must be balanced. In particular, our result shows that
recently proved differential entropy inequalities by Kontoyiannis and Madiman
\cite{KM14} can be deduced from their discrete counterparts due to Tao
\cite{Tao10} in a unified manner. Generalizations to certain abelian groups are
also obtained.
  Our proof of extending inequalities of Shannon entropy to differential
entropy relies on a result of R\'enyi \cite{Renyi59} which relates the Shannon
entropy of a finely discretized random variable to its differential entropy and
also helps in establishing the entropy of the sum of quantized random variables
is asymptotically equal to that of the quantized sum; the converse uses the
asymptotics of the differential entropy of convolutions with weak additive
noise.
