We study a continuous-time pump scheduling problem for a flow transmission task. A finite table of empirical operating points of pump combinations is given, each point specifying a flow rate and power consumption. Electricity prices follow a time-of-use tariff, and combination changes are penalized or limited by per-shift switch caps. We prove a structural theorem: after partitioning time into maximal intervals on which both the tariff and the shift label are the same, called atoms, there exists an optimal schedule in which all switches occur at atom boundaries except possibly one internal switch in one atom. If the selected optimal schedule terminates strictly inside an atom, then no internal atom switch is needed at all. This structure yields an exact atom-level mixed integer linear program whose size depends on the number of atoms rather than on a fine time grid.