Ah, of course! This reminds me of Dirac's delta function, which, in the end, we always integrate. I remember the Dirac delta is also introduced in electrodynamics (cf. Griffiths' Introduction to Electrodynamics (3e) p. 50) to cater for the problem with the divergence of the E-field at r = 0...
In textbooks or other texts that discuss electrodynamics at some point always the term 'uniform' is introduced to describe particularly simple (symmetric) charge distributions, electric fields, etc.
In trying to imagine what a uniform charge distribution - which, of course, is a mathematical...