  The fundamental gap of an interacting many-electron system is given by the
sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The
latter can be generated by advanced approximations to the exchange-correlation
(XC) energy and is the key quantity to capture strong correlation with density
functional theory (DFT). In this work we derive an expression for the
derivative discontinuity in terms of the XC kernel of time-dependent density
functional theory and demonstrate the crucial role of a discontinuity in the XC
kernel itself. By relating approximate XC kernels to approximate local vertex
corrections we then generate beyond-GW self-energies that include a
discontinuity in the local vertex function. The quantitative importance of this
result is illustrated with a numerical study of the local exchange vertex on
model systems.
