This paper investigates the optimal bidding strategy ( supply function) for a generator offering power into a wholesale electricity market. The model has three characteristics: the uncertainties facing the generator are described by a single probability function, namely the market distribution function; the supply function to be chosen is nondecreasing but need not be smooth; the objective function is the expected profit which can be formulated as a Stieltjes integral along the generator's supply curve. In previous work the market distribution function has been assumed smooth, but in practice this assumption may not be satisfied. This paper focuses on the case that the market distribution function is not continuous, and hence an optimal supply function may not exist. We consider a modified optimization problem and show the existence of an optimal solution for this problem. Then we show constructively how such an optimum can be approximated with an epsilon-optimal supply function by undercutting when the generator does not hold a hedging contract ( and possibly overcutting when the generator has a hedging contract). Our results substantially extend previous work on the market distribution model.