  We show weak convergence of quantile and expectile processes to Gaussian
limit processes in the space of bounded functions endowed with an appropriate
semimetric which is based on the concepts of epi- and hypo convergence as
introduced in \citet{buecher2014}. We impose assumptions for which it is known
that weak convergence with respect to the supremum norm or the Skorodhod metric
generally fails to hold. For expectiles, we only require a distribution with
finite second moment but no further smoothness properties of distribution
function, for quantiles, the distribution is assumed to be absolutely
continuous with a version of its Lebesgue density which is strictly positive
and has left- and right-sided limits. We also show consistency of the bootstrap
for this mode of convergence.
