Is dx ever truly equal to zero in integration theory?

[WWGD]

Oh God. What did I write I corrected it.

I just emailed it to your advisor. Please don't email my gross mistakes to anyone :).

 

[WWGD]

Re dx=0 , since I am not fully clear of the context, if you mean in the way it is used in integration theory, then the answer is no;
actually, for a _ fixed Riemann sum_, dx is defined as the least value $x_k -x_{k-1}$ , where the $x_k$ are part of a partition of an interval (integrals on unbounded interval are a separate issue), and it takes on a well-defined minimum value.
For the actual Riemann integral, you consider the _limit_ as $dx \rightarrow 0$ , bt dx is itself not 0, at least not so in this sense of the word.