First, if there exist such an "f" then we must have the mixed derivatives the same. That works for "xy" but
That is, there is a incorrect sign so there is NO SUCH f.
I am going to assume that you meant
The concept is the same as in two dimensions, there is just one more equation. If then we must have , , .
Starting from and integrating, . The "constant of integration" can be any function of y and z. Differentiating that with respect to y, so that . Since g(y, z) is a function of y and z, that "constant of integration" can be a function of z but not x or y.
Now, we have that . Differentiate that with respect to z: . That just says that h'(z)= 0 so that h is a constant: .