I recently came across dirac delta function and trying to learn about it has led me to learn that it is a distribution/generalized function and is not a ordinary function. But most of the explanations as mathematically abstract. So I am trying to understand it using examples. The common example given while explaining the dirac delta function is that it helps express in a mathematically-correct form such idealized concepts as the density of a material point, a point charge or a point dipole, the (space) density of a simple or double layer, the intensity of an instantaneous source, etc. I am wondering how the distribution helps mathematically represent concept such as density. The sentence "On the other hand, the concept of a generalized function reflects the fact that in reality a physical quantity cannot be measured at a point; only its mean values over sufficiently small neighbourhoods of a given point can be measured. Thus, the technique of generalized functions serves as a convenient and adequate apparatus for describing the distributions of various physical quantities. Hence generalized functions are also called distributions." (Ref:https://www.encyclopediaofmath.org/index.php/Generalized_function) means that the mass is "continuous" (in continuum approximation sense), but when talking about "density at a point", in reality we consider the only the mass closely "distributed" around that point. This is why density is a "distribution".
a)So is density not continuous?
b)How does the integral: $$ \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) $$ help explain the concept of density? is the test function $f(x)$ giving the continuous density through out the one dimensional space and the delta function help us get the density at a point a?
C)Why not directly get the value of the trial function at point 'a' using f(a) instead of using the round about way of delta fucntion?
Note: Here is a good notes that I found useful -> https://math.mit.edu/~stevenj/18.303/delta-notes.pdf