The electric potential must be continuous, otherwise the electric field will be infinity (E=-dv/dx).
But, is there a a physical law that requireing the electrochemical potential to be continuous?
Dear Gotleyb,
Your question is very interesting. I've asked the same question to my teachers during my doctorate and nobody could respond it. And you're right: changes in electric field in electrochemistry changes the electrochemical systems
On the other hand, we have to think about the electrochemical system as a whole: the electric field will induce the ion's mobilities in solution, and the double layer will change its composition. Although there's no law in electrochemistry stating that electrical fiel should be constant, we use a supporting electrolyte to make it constant between electrodes, and the transport number for analyte is so small if compared with electrolyte's transport number, and this analyte does not suffer any influence from electric field.
We have also another question: we cannot assume a electrostatic arguments for electrochemistry (even from steady state experiments such impedance spectroscopy and potentiometry) because the electrical charges in all electrode changes all the time. Maybe some arguments from electrodynamics can help.