What should be continuous at the interface of two materials, electric potential chemical potential or electrochemical potential? At the interface between:
1) conductor/conductor
2) conductor/semiconductor (or dielectric)
3) semiconductor/semiconductor (or dielectric/dielectric)
What quantity should be continuous?
Is it the electrochemical potential, only the chemical potential or is it the electric potential?
Since they are all related by $V_{ec}=V_c+V_e$, if two of them are continuous then all of them are continuous, but I think this situations cannot be true.
However, I know that the Fermi energy is continuous (which I think is the chemical potential but I'm not sure since there's a lot of misleading information). And also the electric potential should be continuous, otherwise the electric field (defined by $E=-grad(V_e)$) would be infinity.
The question refers both for equilibrium and for an applied voltage (steady currents).
I couldn't find any information on this matter. All books on semiconductors or electrostatic/dynamic does not mention it and does not get to much deep into such physics.
 A: I think the answer depends on whether the system is at thermodynamics equilibrium rather than on the type of material.
In the case of out of thermodynamics equilibrium, the chemical potential need not be continuous. If it has different values in both materials, then there will be a flow of matter (i.e. electrons) until thermodynamics equilibrium is reached, if ever.
If the system is at thermodynamics equilibrium then the chemical potential will be equal in both materials, and thus continuous at their interface.
Regarding the comment of rob: When an electric field is applied, the current is driven by both this field plus the gradient of the chemical potential. So a voltmeter measures the electrochemical potential, not the electric potential. Maybe this is a starting point to understand thermocouples.
About your doubts on the Fermi energy, I suggest checking Wikipedia's article (https://en.wikipedia.org/wiki/Fermi_energy). In short: No, the Fermi energy is not equal to the chemical potential although in metals one can think of them being nearly equal. But in reality, Fermi energy is only defined at absolute 0 temperature, while the latter is defined for any temperature. Another useful reference is "Introduction to Solid State Physics" page 573 by Ashcroft and Mermin. 
Lastly, it seems you take for granted that $\vec E$ must not diverge, while if you take the example of a point charge at a particular location, $\vec E$ is precisely undefined (or infinite) at that particular position.
