  Training of conditional WGANs is usually done by averaging the underlying
loss over the condition. Depending on the way this is motivated different
constraints on the Lipschitz continuity of the discriminator arise. For the
weaker requirement on the regularity there is however so far no mathematically
complete justification for the used loss function. This short mathematical note
intends to fill this gap and provides the mathematical rationale for
discriminators that are only partially Lipschitz-1 for cases where this
approach is more appropriate or successful.
