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$\begingroup$ Mathematically speaking, $dq$ refers to a measure on the position space. This measure can incorporate continuous and discrete charges simultaneously. I agree that practically it is not very useful but IMHO it is a convenient shorthand that directly conveys the idea behind the formula. Dirac delta is only necessary when you want to change the measure from $dq \to dr$. $\endgroup$
– SmallPieceOfBread
Commented Jun 13 at 21:30

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