Dirac delta function is continuous and differential

[astro2cosmos]

since dirac delta function is not a literally a function but a limit of function,does it mean that dirac delta function is continuous and differentiable through out the infinity?
is there any example of dirac delta function if yes then give meeeeeeee?

 

[DrFaustus]

astro2cosmos -> Strictly speaking, the Dirac delta is a distribution, that is it's a functional on the space of smooth and compactly supported functions. As such, one has to be a bit careful as to what does it mean for it to be "differentiable" and "continuous". Still, it is both differentiable and continuous. (I'm sure about differentiability, but have a doubt about continuity...)

is there any example of dirac delta function if yes then give meeeeeeee?

What is that supposed to mean?

 

[madness]

I don't know how to write in latex but examples of dirac delta are the top hat function as the width goes to zero and the (properly normalised) gaussian as the width goes to zero.

 

[friend]

since dirac delta function is not a literally a function but a limit of function,does it mean that dirac delta function is continuous and differentiable through out the infinity?
is there any example of dirac delta function if yes then give meeeeeeee?

The internal parameter that goes to zero inside the dirac delta is INDEPENDENT of the variable parameter that goes to zero in the calculus process. For any given value of the internal parameter (it is never exactly zero), the dirac delta function is continuous, differentialbe, and integrable as far as calculus is concerned.

 

[madness]

Here's what I learned in my course. There are several functions which in a certain limit approach the dirac delta. The examples I gave are the top hat and the gaussian. The top hat is not differentiable, but the gaussian is. I don't know much more than that.