reference post 1.
Functions defined on a closed region can only be continuous and have continuous derivatives at the boundary if they are defined beyond the closed region. After all operations are performed and solutions found, the functions outside the closed region can be ignored if they are irrelevant.
The essence of the general problem condenses to a consideration of three common versions of “The Fundamental Theorem of Calculus.”
I) If f is continuous on [a,b] it has an anti-derivative (F’(x)=f(x)) on [a,b].
II) If f is continuous on (a,b) it has an anti-derivative (F’(x)=f(x)) on (a,b).
III) If f is continuous on (c,d) and (a,b) is in (c,d), f has an antiderivative on (a,b).
I) & II) generalized is the case for generalized divergence theorems and differential equations defined on a closed “volume” with boundary conditions, but in these cases the functions and derivatives can’t be continuous on the “surface” of the closed “volume.”
It turns out that left and right sided functions work even though they don’t satisfy the requirements of continuity as explained in post 1 above:
One sided functions are tangents at the boundary which implies the function could be extended slightly beyond the boundary along the tangent “plane” or normal to the tangent (directional derivative) which turns cases I) and II) into case III. Same for higher order derivatives with function at boundary extended in such a way as to make the higher order derivatives exist and be continuous at boundary.
So cases I and II turn out to be right if you explain why.
Textbook authors don’t explain it. They use the general version of I) or II). I give Kellog’s (Potential Theory) version of the Divergence Theorem because it is typical and uses cases I) and II) (generalized), and some authors defer to his derivation.
“The divergence theorem involves two things, a certain region, or portion of space, and a vector field, or set of three functions X, Y, Z of x,y,z, defined in this region.”….. “As to the field (X,Y,Z), we shall assume that its components and partial derivatives of the first order are continuous within and on the boundary of N.”