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In this paper, the Riemann problem for the pressureless Euler equations witha discontinuous source term is considered. The delta shock wave solution isobtained by combining the generalized Rankine-Hugoniot conditions together withthe method of characteristics for different situations, which reflects theimpact of the source term on the delta shock front. Moreover, during theconstruction process of the Riemann solution, some interesting phenomena arealso observed, such as the disappearance of the delta shock wave and theoccurrence of the vacuum state, etc.

Singular solutions to the Riemann problem for the pressureless Euler equations with discontinuous source term