Distribution Theory Formulation
The theory of distributions eliminates analytic problems with the symmetry. The derivative of any integrable function can be defined as a distribution. The use of integration by parts puts the symmetry question back onto the test functions, which are smooth and certainly satisfy the symmetry. In the sense of distributions, symmetry always holds.
Another approach, which defines the Fourier transform of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously.
Read more about this topic: Symmetry Of Second Derivatives
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