Multi Switch RC Circuit Analysis
At t=0 ms, you are correct, constant 4V from the voltage divider
$$ \frac{R_3}{R_1+R_2+R_3} V_1 = \frac{4}{10} (10 \text{V})= 4 \text{V}$$

At t=0.1 ms, switch 1 closes, while switch 3 remains closed, so C3 is now in parallel with R3. Your question is actually ill-posed because it doesn't state how much C3 was charged, but I will assume that it is discharged and thus 0V at \$t=0.1 \text{ms}\$. (Capacitors resists voltage change, so initially, the voltage across the capacitor has to be the same as what it was before it switched.)

If so, when the switch closes, all current will go towards charging the capacitor. Since then there is no current going through R3, the voltage will be 0V. If that was the state of things, it would continue charging the capacitor until no current goes into the capacitor, and we're back to 4V.

If you are worried about the discontinuity, then don't. In a real circuit, there is inductance due to the closed loop of the circuit itself, so the current change over R3 is not instantaneous, only "very fast".