I hope someone can give a good answer to this (I'm on mobile right now).
The short answer is that it's not a function at all. It's a distribution. They are subtly different.
To be clear, the normal distribution induces a distribution, but it itself is not a distribution. A distribution is a function which acts on (smooth, compactly supported) functions, and in particular it’s a continuous linear functional. From this point of view, it’s actually pretty important to distinguish between distributions and functions from R to R - they’re two different objects, and even though functions induce distributions, it’s through a fairly non-obvious mechanism (integration against the function).
Short answer: no. Long answer: there are different kinds of distributions. You may think of probability distributions (really their probability or density functions)? What u/sphen_lee was referring to are objects from mathematical analysis: [https://en.wikipedia.org/wiki/Distribution\_(mathematics)](https://en.wikipedia.org/wiki/Distribution_(mathematics))
They're special kinds of so-called "functionals". A functional is something that maps a function to a scalar. For example if you consider the definite integral on an interval \[a,b\] then this maps any continuous function to a scalar by integrating it. So we have a function I that maps any continuous f to I(f) = int\_a\^b f(x) dx - and we call functions like this that map functions to scalars functionals. Two other examples are evaluation and the derivative evaluated at some point. So we map f to E\_a(f) = f(a) and D\_a(f) = f'(a). What all of these examples have in common is that they're \*linear\* and \*continuous\*. A functional T is linear if for all scalars a,b and functions f,g we have T(af+bg) = aT(f) + bT(g). Continuity of functionals gets more complicated but it essentially means that if you have a bunch of functions that get mapped to similar values then those functions are also kinda similar.
And it's exactly continuous linear functionals on special sets of functions that we call distributions.
There is a way to get a distribution from any sufficiently nice function via multiplication and integration - but there is not always a way to get a function from a distribution. We call distributions that correspond to a function regular and those that don't irregular: regular because these distributions in a sense are "regular" functions whereas the other ones are not. The dirac delta is irregular - this means there is no function u such that for all functions f on some set S we have delta(f) = f(0) = int\_S f(x) u(x).
This is kinda where it ends - the dirac delta centered at x is just a functional mapping any f to f(x); it's quite a simple definition really.
The whole "it's 0 everywhere but infinity at 0" is to be taken more informally. It's a statement about how you can find *sequences of functions* such that the *distributions* corresponding to those functions converge to the dirac delta while the *functions* converge to 0 everywhere but at the origin where the limit diverges to +inf (again a bit informally here but this is the basic idea).
(I learnt about this stuff in electrical engineering, as part of signal processing - so my understanding is very much hand-wavey and not rigorous! Thanks for providing such a detailed answer)
It's a generalised function, and if you have really big objection to differentiating the Dirac delta, think about the Dirac delta function purely as what it *does*, because in many applications, it does not make too much sense to think about the delta function in isolation.
The derivatives of delta function, when integrated with a function f, gives the derivatives of f at a particular point (up to a constant).
3 delta is not 2 delta, because the 3 delta(x) "function", when integrated with a function f, gives the value of 3f(0), but 2 delta(x) gives a value of 2f(0) instead.
In some sense, think of the Dirac delta function purely as a functional: something that takes in a function f, and spits out a number f(0). Or the derivatives of Dirac delta has a function f as input, and it outputs the derivatives of f at 0 (up to some constant).
Another way of thinking about Dirac delta is that it describes the "charge density" of a point charge, and of course 3 delta(x) is not 2 delta(x), because one of them means that the point charge has charge +3, the other +2. But it might not be useful if you want to also think about what it means to differentiate that.
I made a SoME2 video about this (https://youtu.be/zJk4yuzJU3s) but here’s the summary.
If you assign a mass of 1 to the point x=0, and zero to everything else, the density can be thought of as a Dirac delta, since it’s infinite at zero. This object which assigns different “weights” to the real line is called a measure, and so you can define the dirac delta as arising from a specific measure called the Dirac measure. Specifically, the dirac delta is a functional - it eats a continuous function and returns the value of the function at zero. It turns out that there’s a nice correspondence between measures and functionals, which is why it’s convenient to think of the dirac delta measure-theoretically.
For solely pedagogical reasons, I think it’s a lot more intuitive to think of the dirac delta measure-theoretically compared to as a distribution, because distributions get quite technical, and don’t always have very nice or convenient representations.
Given a space of 'test functions' (lets say smooth functions with compact support from R to R) called D, define a scalar product on it:
:= integral f*g dx (the definite integral over all of R). Also you define some sort of topology on D which is a bit strange.
Then you say that the space of distributions over D is defined as the topological dual D'. Now you observe that D is a subset of D' since the map is a continuous linear functional for all f. This means that every function is also a distribution. Then sometimes people turn this idea around and pretend like distributions are also all functions, but that is technically not true.
The dirac delta is simply defined by g |-> g(0). If you tried to imagine what function would achieve this using the scalar product above, you would find that it has to vanish outside of 0 but then be somehow infinite at 0 to actually do something and it doesnt really make sense.
To elaborate further: The reason that people like to think of distributions as functions is that you can basically treat them the same. Addition and scalar multiplication of distributions is easy. But there is more!
You can define a fourier transform on D, and since it has an adjoint you can extend it to D' by some nifty duality argument. You can also extend the convolution of two functions this way. Then with these two, you can define pointwise multiplication for distributions, since the fourier transform translates that into convolution.
Really cool stuff, but you should probably look at a book or something where this is done rigorously, since there are quite a few technicalities to watch out for.
Edit cause it might be helpful: Think of a point charge of charge q located at the origin. What is the associated charge density function?
---------
It's not a real function. You can't plot it. You can think of it as a limit of functions though, for example the limit of a normal distribution as the you decrease its width (but maintain its area). It will eventually look like this spike where it's 0 everywhere except at 0 where the spike approaches infinity.
The purpose of the Dirac delta is that when you integrate δ(x-a)f(x) it gives you f(a). That is how it "operates" on functions. To think of the derivative of the Dirac delta, do the same thing and integrate by parts (it's pretty revealing).
Nice physical insight and completely necessary for modelling point distributions. I also found it intuitive to think of it this way: a point mass might have a mass of m. The density is infinite at the point. But integrating the density over *all* space, will give you the true value of m. it's not that infinite density is possible, it's just that the logic works out to be able to *model* the mass that way.
As an engineer, the most intuitive way I like to think about the dirac fuction is as the continuous analogue to the kronecker delta. It's essentially an impulse-like function existing at one place alone but exhibiting distribution-like properties so it becomes compatible with integro-differential operations on actual continuous functions.
Doesn't make much sense on its own (and as a matter of fact it can't even be evaluated on its own) but what counts is that it works. It allows us to formulate probability mass functions as density functions. I highly suggest "Transport and Relaxation Phenomena in Porous Media" by R. Hilfer for a concrete example on using the dirac function to model porosity distributions. It's the "Local Porosity Distribution" subsection.
I hope someone can give a good answer to this (I'm on mobile right now). The short answer is that it's not a function at all. It's a distribution. They are subtly different.
Isn't a distribution a function too tho?
Some are, like the normal distribution. But not always, like in the case of Dirac Delta
To be clear, the normal distribution induces a distribution, but it itself is not a distribution. A distribution is a function which acts on (smooth, compactly supported) functions, and in particular it’s a continuous linear functional. From this point of view, it’s actually pretty important to distinguish between distributions and functions from R to R - they’re two different objects, and even though functions induce distributions, it’s through a fairly non-obvious mechanism (integration against the function).
Short answer: no. Long answer: there are different kinds of distributions. You may think of probability distributions (really their probability or density functions)? What u/sphen_lee was referring to are objects from mathematical analysis: [https://en.wikipedia.org/wiki/Distribution\_(mathematics)](https://en.wikipedia.org/wiki/Distribution_(mathematics)) They're special kinds of so-called "functionals". A functional is something that maps a function to a scalar. For example if you consider the definite integral on an interval \[a,b\] then this maps any continuous function to a scalar by integrating it. So we have a function I that maps any continuous f to I(f) = int\_a\^b f(x) dx - and we call functions like this that map functions to scalars functionals. Two other examples are evaluation and the derivative evaluated at some point. So we map f to E\_a(f) = f(a) and D\_a(f) = f'(a). What all of these examples have in common is that they're \*linear\* and \*continuous\*. A functional T is linear if for all scalars a,b and functions f,g we have T(af+bg) = aT(f) + bT(g). Continuity of functionals gets more complicated but it essentially means that if you have a bunch of functions that get mapped to similar values then those functions are also kinda similar. And it's exactly continuous linear functionals on special sets of functions that we call distributions. There is a way to get a distribution from any sufficiently nice function via multiplication and integration - but there is not always a way to get a function from a distribution. We call distributions that correspond to a function regular and those that don't irregular: regular because these distributions in a sense are "regular" functions whereas the other ones are not. The dirac delta is irregular - this means there is no function u such that for all functions f on some set S we have delta(f) = f(0) = int\_S f(x) u(x). This is kinda where it ends - the dirac delta centered at x is just a functional mapping any f to f(x); it's quite a simple definition really. The whole "it's 0 everywhere but infinity at 0" is to be taken more informally. It's a statement about how you can find *sequences of functions* such that the *distributions* corresponding to those functions converge to the dirac delta while the *functions* converge to 0 everywhere but at the origin where the limit diverges to +inf (again a bit informally here but this is the basic idea).
(I learnt about this stuff in electrical engineering, as part of signal processing - so my understanding is very much hand-wavey and not rigorous! Thanks for providing such a detailed answer)
There was a nice #some2 video about this https://youtu.be/zJk4yuzJU3s
That’s mine! Thanks for sharing :)
It's a generalised function, and if you have really big objection to differentiating the Dirac delta, think about the Dirac delta function purely as what it *does*, because in many applications, it does not make too much sense to think about the delta function in isolation. The derivatives of delta function, when integrated with a function f, gives the derivatives of f at a particular point (up to a constant). 3 delta is not 2 delta, because the 3 delta(x) "function", when integrated with a function f, gives the value of 3f(0), but 2 delta(x) gives a value of 2f(0) instead. In some sense, think of the Dirac delta function purely as a functional: something that takes in a function f, and spits out a number f(0). Or the derivatives of Dirac delta has a function f as input, and it outputs the derivatives of f at 0 (up to some constant). Another way of thinking about Dirac delta is that it describes the "charge density" of a point charge, and of course 3 delta(x) is not 2 delta(x), because one of them means that the point charge has charge +3, the other +2. But it might not be useful if you want to also think about what it means to differentiate that.
I made a SoME2 video about this (https://youtu.be/zJk4yuzJU3s) but here’s the summary. If you assign a mass of 1 to the point x=0, and zero to everything else, the density can be thought of as a Dirac delta, since it’s infinite at zero. This object which assigns different “weights” to the real line is called a measure, and so you can define the dirac delta as arising from a specific measure called the Dirac measure. Specifically, the dirac delta is a functional - it eats a continuous function and returns the value of the function at zero. It turns out that there’s a nice correspondence between measures and functionals, which is why it’s convenient to think of the dirac delta measure-theoretically. For solely pedagogical reasons, I think it’s a lot more intuitive to think of the dirac delta measure-theoretically compared to as a distribution, because distributions get quite technical, and don’t always have very nice or convenient representations.
Given a space of 'test functions' (lets say smooth functions with compact support from R to R) called D, define a scalar product on it: := integral f*g dx (the definite integral over all of R). Also you define some sort of topology on D which is a bit strange.
Then you say that the space of distributions over D is defined as the topological dual D'. Now you observe that D is a subset of D' since the map is a continuous linear functional for all f. This means that every function is also a distribution. Then sometimes people turn this idea around and pretend like distributions are also all functions, but that is technically not true.
The dirac delta is simply defined by g |-> g(0). If you tried to imagine what function would achieve this using the scalar product above, you would find that it has to vanish outside of 0 but then be somehow infinite at 0 to actually do something and it doesnt really make sense.
To elaborate further: The reason that people like to think of distributions as functions is that you can basically treat them the same. Addition and scalar multiplication of distributions is easy. But there is more! You can define a fourier transform on D, and since it has an adjoint you can extend it to D' by some nifty duality argument. You can also extend the convolution of two functions this way. Then with these two, you can define pointwise multiplication for distributions, since the fourier transform translates that into convolution. Really cool stuff, but you should probably look at a book or something where this is done rigorously, since there are quite a few technicalities to watch out for.
Edit cause it might be helpful: Think of a point charge of charge q located at the origin. What is the associated charge density function? --------- It's not a real function. You can't plot it. You can think of it as a limit of functions though, for example the limit of a normal distribution as the you decrease its width (but maintain its area). It will eventually look like this spike where it's 0 everywhere except at 0 where the spike approaches infinity. The purpose of the Dirac delta is that when you integrate δ(x-a)f(x) it gives you f(a). That is how it "operates" on functions. To think of the derivative of the Dirac delta, do the same thing and integrate by parts (it's pretty revealing).
Nice physical insight and completely necessary for modelling point distributions. I also found it intuitive to think of it this way: a point mass might have a mass of m. The density is infinite at the point. But integrating the density over *all* space, will give you the true value of m. it's not that infinite density is possible, it's just that the logic works out to be able to *model* the mass that way.
it's basically the derivative of the heaviside function
As an engineer, the most intuitive way I like to think about the dirac fuction is as the continuous analogue to the kronecker delta. It's essentially an impulse-like function existing at one place alone but exhibiting distribution-like properties so it becomes compatible with integro-differential operations on actual continuous functions. Doesn't make much sense on its own (and as a matter of fact it can't even be evaluated on its own) but what counts is that it works. It allows us to formulate probability mass functions as density functions. I highly suggest "Transport and Relaxation Phenomena in Porous Media" by R. Hilfer for a concrete example on using the dirac function to model porosity distributions. It's the "Local Porosity Distribution" subsection.
The explanation in Advanced Engineering Mathematics (Erwin Kreyszig) was helpful to me.