A quantum point contact (QPC) patterned on a two-dimensional electron gas is investigated with a scanning gate setup operated at a temperature of 300 mK. The conductance of the point contact is recorded while the local potential is modified by scanning the tip. Discrete charging of traps induced by the local potential is observed as a stepwise conductance change of the constriction. By selectively changing the state of some of these traps, it is possible to observe changes in the transmission of the QPC. The lateral position of such traps is determined, and their density is estimated to be below 50 per mu m(2), corresponding to less than 1% of the doping concentration.