Indifference curve - Does $dU = 0$ hold in higher dimensions? / Problem of integrability
Seems to me $dU = 0$ is true by definition unless $dU$ is not defined 


because the function $U$ is not differentiable in some variable. 
E.g. $$U(x,y) = |x| + y$$
because there are a countably infinite number of dimensions but the partial differentials are not always absolute convergent and hence cannot be summed up. E.g.: $$U(x_1,x_2,x_3,x_4...) = x_1 - x_2 + x_3 -x_4...$$
because the dimensions have a larger than countably infinite cardinality. I have never seen this in micro but it is theoretically possible. E.g. expected utility over continuum states.


Edit: 
This seems to offer an extremely thorough walkthrough of the integrability problem in $n$ dimensions.