  Matthias Schr\"oder has asked the question whether there is a weakest
discontinuous problem in the continuous version of the Weihrauch lattice. Such
a problem can be considered as the weakest unsolvable problem. We introduce the
discontinuity problem, and we show that it is reducible exactly to the
effectively discontinuous problems, defined in a suitable way. However, in
which sense this answers Schr\"oder's question sensitively depends on the
axiomatic framework that is chosen, and it is a positive answer if we work in
Zermelo-Fraenkel set theory with dependent choice and the axiom of determinacy
AD. On the other hand, using the full axiom of choice, one can construct
problems which are discontinuous, but not effectively so. Hence, the exact
situation at the bottom of the Weihrauch lattice sensitively depends on the
axiomatic setting that we choose. We prove our result using a variant of Wadge
games for mathematical problems. While the existence of a winning strategy for
player II characterizes continuity of the problem (as already shown by Nobrega
and Pauly), the existence of a winning strategy for player I characterizes
effective discontinuity of the problem. By Weihrauch determinacy we understand
the condition that every problem is either continuous or effectively
discontinuous. This notion of determinacy is a fairly strong notion, as it is
not only implied by the axiom of determinacy AD, but it also implies Wadge
determinacy. We close with a brief discussion of generalized notions of
productivity.
