This work is motivated by a quite general question: Under which circumstances are the capacities of information transmission systems continuous? The research is explicitly carried out on arbitrarily varying quantum channels (AVQCs). We give an explicit example that answers the recent question whether the transmission of messages over AVQCs can benefit from distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the deterministic capacity does have discontinuity points, while that behaviour is, at the same time, not generic: We show that it is continuous around its positivity points. This is in stark contrast to the randomness-assisted capacity, which is always continuous in the channel. Our results imply that the deterministic message transmission capacity of an AVQC can be discontinuous only in points where it is zero, while the randomness assisted capacity is nonzero. Apart from the zero-error capacities, this is the first result that shows a discontinuity of a capacity for a large class of quantum channels. The continuity of the respective capacity for memoryless quantum channels had, among others, been listed as an open problem on the problem page of the ITP Hannover for about six years before it was proven to be continuous. We also quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) decoding error with respect to the average error criterion that can be achieved over a finite number of channel uses and the number of messages that can be sent. This part of our results also applies to entanglement- and strong subspace transmission. In addition, we give a new sufficient criterion for the entanglement transmission capacity with randomness assistance to vanish.