For continuous functions of several variable, by Young's theorem the order of integration (i.e. "antiderivation") does not matter so that for instance
$$ \int \left(\int f(x,y)~ dx\right) ~dy = \int \left(\int f(x,y) ~dy\right) ~dx $$
Now we also know that some non-continuous functions have antiderivatives (see the wikipedia article on antiderivative).
My question is :
- Is it also true in general that the order of integration does not matter for these non-continuous functions which have antiderivative?
- Does anyone know of a proof?
- If not true in general, does there exists additional conditions such that the result would hold for non-continuous functions?
This question emerge from spliting this question into two because I thought it was too messy.