  We study the loss landscape of training problems for deep artificial neural
networks with a one-dimensional real output whose activation functions contain
an affine segment and whose hidden layers have width at least two. It is shown
that such problems possess a continuum of spurious (i.e., not globally optimal)
local minima for all target functions that are not affine. In contrast to
previous works, our analysis covers all sampling and parameterization regimes,
general differentiable loss functions, arbitrary continuous nonpolynomial
activation functions, and both the finite- and infinite-dimensional setting. It
is further shown that the appearance of the spurious local minima in the
considered training problems is a direct consequence of the universal
approximation theorem and that the underlying mechanisms also cause, e.g.,
$L^p$-best approximation problems to be ill-posed in the sense of Hadamard for
all networks that do not have a dense image. The latter result also holds
without the assumption of local affine linearity and without any conditions on
the hidden layers.
