  We prove that a broad array of capacities of a quantum channel are
continuous. That is, two channels that are close with respect to the diamond
norm have correspondingly similar communication capabilities. We first show
that the classical capacity, quantum capacity, and private classical capacity
are continuous, with the variation on arguments epsilon apart bounded by a
simple function of epsilon and the channel's output dimension. Our main tool is
an upper bound of the variation of output entropies of many copies of two
nearby channels given the same initial state; the bound is linear in the number
of copies. Our second proof is concerned with the quantum capacities in the
presence of free backward or two-way public classical communication. These
capacities are proved continuous on the interior of the set of non-zero
capacity channels by considering mutual simulation between similar channels.
