Q:

Can an electronic device continuously sweep the voltage?

Modern power supplies can sweep the potential in a given range with a specific rate. I am curious if the electronic devices can continuously sweep the voltage (in a perfect sense) or it is just step-based changes, which seem like sweeping?
For example, consider sweeping with from 0 to 10V in 10,000s. Does the sweep has a resolution of say 1mV, and the data looks like
time/s    voltage/V
1         0.001
2         0.002
3         0.003

Or it is absolutely continuous, as the voltage is 0.0025 at 2.5s, 0.002513 at 2.513s?
Note that my question is not about the voltage resolution, but if the change from 2 and 3s is a step from 0.002 to 0.003V or it is still a gradual change.

A:

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.

A:

It really depends. 
If the sweep is generated digitally by a micro then no. 
If it's generated using a saw-tooth curve based on charging some cap then much more so. 
However, nothing is truly linear, and arguably if it is a switching power supply there are a few more variables you need to worry about.

A:

It depends really much on the PSU. 

If a R2/R ladder is used, then it will move in steps, but right at the step-transition it will be charging/discharging some parasitic capacitance so it won't be a true discontinued step. It will still be connected. But you can put as many stages as you want in your R2/R ladder network to improve on your resolution, but the more stages you add the more sensitive it will be to noise, because you're making the bits mean less and less and less the longer the ladder gets. 

If however the sweep is made by using a Constant Current Source together with a capacitor, then you will see the voltage as a straight line. But at what resolution though? If we look at the picosecond scale then we will see the steps yet again as electrons move around. Same thing with the R2/R ladder, though it's much easier to notice. 
A MOSFET can be utilized to make a Voltage Controlled Current Source, and it can be controlled in a digital manner to set the \$\frac{dV}{dT}\$. You can set the voltage of the gate with an R2/R ladder, then the straight line (Voltage across the Capacitor over Time) will have different finite angles. Or you could use an op-amp together with an R2/R ladder and make use of control theory to make the op-amp come up with the voltage for the gate that will make the capacitor charge at some specific \$\frac{dV}{dT}\$ so it takes 10 seconds for it to reach X volt. It's still digital, though the op-amp does all the magic. 

Or just put a motor on a potentiometer. Again, it depends on the PSU you're talking about. An answer is ever as good as the question, a broad question gives an uncertain and broad answer. 

