  The regularity of images generated by convolutional neural networks, such as
the U-net, generative networks, or the deep image prior, is analyzed. In a
resolution-independent, infinite dimensional setting, it is shown that such
images, represented as functions, are always continuous and, in some
circumstances, even continuously differentiable, contradicting the widely
accepted modeling of sharp edges in images via jump discontinuities. While such
statements require an infinite dimensional setting, the connection to
(discretized) neural networks used in practice is made by considering the limit
as the resolution approaches infinity. As practical consequence, the results of
this paper in particular provide analytical evidence that basic L2
regularization of network weights might lead to over-smoothed outputs.
