We show that, even for monotone directionally differentiable Lipschitz func
tionals on Hilbert spaces, basic concepts of generalized derivatives identi
fy only particular pseudo regular (or metrically regular) situations. Thus,
pseudo regularity of (multi-) functions will be investigated by other mean
s, namely in terms of the possible inverse functions. Tn this way, we show
how pseudo regularity for the intersection of multifunctions can be directl
y characterized and estimated under general settings and how contingent and
coderivatives may be modified to obtain sharper regularity conditions. Con
sequences for a concept of stationary points as limits of Ekeland points in
nonsmooth optimization will be studied, too.