We obtain the first tight continuityestimate on the Shannon entropy of random variables with a countably infinitealphabet . The proof relies on a new mean-constrained Fano-type inequality and the notion of maximal coupling . The above scheme works only for Shannon- and von Neumann entropies . We develop a novel approximation scheme whichrelies on recent results on operator H\”older continuous functions and theequivalence of all Schatten norms in special spectral subspaces of the Hamiltonian . Finally, we settle an open problem on related approximationquestions posed in the recent works by Shirokov on the so-calledFinite-dimensional Approximation (FA) property . We prove a variety of new and refined uniform continuity bounds for entroves of both classical random variables on an infinite state space and of quantumstates of infinite-dimensional systems. Bounds for $\alpha>1$ are provided, too. Finally, We settle an answer to related approxumsumsumsentropies. We conclude an open question on related approxuesuesuesesesesions posed on the

Author(s) : Simon Becker, Nilanjana Datta, Michael G. Jabbour