Per se definition of Dirac's Delta funcion

We examine critically the various definitions of the Dirac's Delta, we note that we are constantly in the presence of a discontinuity at infinity which makes it impossible to give a correct definition of the Dirac's Delta per se. Then we give its definition per se according to the standards dictated by Dirac himself: it is a 'function' that is zero everywhere except at the zero point where it has an infinite value and is such that the integral from minus infinity to plus infinity is 1 and that also multiplied by a 'well-behaved function' it gives the value of the function at the point 0. Note also that if we accept the Cesàro summability we arrive quickly, also in this case, at a correct per se definition of the Delta itself. We are then reasoning on the relationship between mathematics and physics.