We consider an infinite horizon discounted cost minimization problem for a one-dimensional stochastic differential equation model. The available control is an added bounded variation process. The cost structure involves a running cost function and a proportional cost for the use of the control process. The running cost function is not necessarily convex. We obtain sufficient conditions to guarantee the optimality of the zero control. Also, for unbounded cost functions, we provide sufficient conditions which make our optimal state process a reflecting diffusion on a compact interval. In both cases, the value function is a C-2 function. For bounded cost functions, under additional assumptions, we obtain a complex optimal strategy which turned out to be a mixture of jumps and local-time-type processes. In this case, we show that the value function is only a C-1 function and that it fails to be a C-2 function. We also discuss a related variance control problem.