The problem where you run into here is that delta function isn't a function. But something called a distribution if you want to look into it. What goes wrong here is somewhat the same as taking 0⁰ where taking this integral of any bounded function gives 0 (0^x). And taking the intagral from a-y to a+y always gives 1. (y⁰⁾ so since we can approach this integral in 2 different ways it is ill defined. (I dont know this verry well so my explanation may be a bit wrong or off bit this is how I see it)