Why is $\delta[an]=\frac 1 a \delta[n]$ in discrete time?

1. Why is the integral of Delta 1?
2. Why is the Dirac delta function not a function?
3. Why do we use Dirac delta function?
4. Is the Dirac delta function continuous?

Why is the integral of Delta 1?

The definition of the delta function is that integrating a “test function” against it is evaluating that test function at 0. If that “test function” is the constant function 1, evaluation of that constant function at 0, or anywhere, is 1.

Why is the Dirac delta function not a function?

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different.

Why do we use Dirac delta function?

The Dirac delta function is an important mathematical object that simplifies calculations required for the studies of electron motion and propagation. It is not really a function but a symbol for physicists and engineers to represent some calculations.

Is the Dirac delta function continuous?

The Dirac delta function, often referred to as the unit impulse or delta function, is the function that defines the idea of a unit impulse in continuous-time. Informally, this function is one that is infinitesimally narrow, infinitely tall, yet integrates to one.