  We study slow variation (both spatial as well as temporal) of a parameter of
a system in the vicinity of discontinuous quantum phase transitions, in
particular, a discontinuity critical point (DCP) (or a first-order critical
point). We obtain the universal scaling relations of the density of defects and
the residual energy after a temporal quench, while we also unravel the scaling
of the characteristic length scale associated with a spatial quench of a
symmetry breaking field. Considering a spin-1/2 XXZ chain we establish how
these scaling relations get modified when the DCP is located at the boundary of
a gapless critical phase; these predictions are also confirmed numerically.
