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
a) If you tried to define what "density" is and what "continuous" for it means, you would find that it makes no sense. As you use it, it defines a measure, that is, it assigns every interval (and thus every Borel set) some positive value. One can show that this is equivalent to a linear continuous functional on the space of continuous test functions, either compactly supported or fast falling. In that later sense, continuity is built-in.
b) It does not explain anything, it is just a definition of the left side by the right, as the left side does not have a meaning in itself. You can use it as another symbolic way to denote the application of the delta distribution to a test function. The non-symbolic content is the right side, it defines what the left side means.
c) You are right, one could say that this is just the evaluation operator in $a$ and write $E_a(f)=f(a)$. But Dirac introduced this notation and since then it is traditional. It is also compatible with the idea of an approximate identity (of the convolution of $L^1$ functions) which approximate delta in a distributional sense. To recap, any function $\phi$ which is non-negative, and integrable with integral $1$ defines an approximate identity (at $x=0$) with the sequence of functions $\phi_n(x)=n\phi(nx)$. So it is common to write $\phi_n\to\delta_0$ for $n\to\infty$ (in the topology of distributions).
See Carl Offner: "A little harmonic analysis" for a deeper trieatment of these ideas.