Can an electronic device continuously sweep the voltage?
If using a digital system, then yes, you would expect a "staircase" of 1 mV steps. This occurs internally in many analog to digital converters. The input is compared with an internal binary-generated staircase waveform and when the input level is reached the binary count is read and stored.



Figure 1. Counter type ADC. Source: Daenotes.

An analog triangle-wave or sawtooth generator would give a continuous sweep.