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A function is a set of ordered pairs, such that no two ordered pairs share the same first entry.

I have heard that distributions are called "generalizations of functions". Simply looking at their definition, I wasn't able to see where they were generalizing the concept of function. I am under the impression that distributions are a kind of continuous linear functional.

I looked online and couldn't find a clear answer (and wasn't too fond of the Wikipedia article).

Are distributions actually different from the modern set-theoretic definition of a function, or is the slogan "generalization of a function" an antiquated or convenient term among analysts? If they truly can be different, then give good example of where the modern function definition fails. Otherwise, a concise "yes" will suffice.

Nate
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    Some people have a limited view. "Objects" are numbers; sets are finite; functions are real-valued; etc. (Not everyone holds all of these views, of course). This comes from research-focus. If you're always writing $\frac1n$, it makes sense to not include $0$ as a natural number (admittedly, of course, that's wrong, $0$ is a natural number). In some contexts "function" is a convenient synonym for "real-valued function". I knew I was going to be a set theorist when I didn't see the problem with a function taking values outside of $\Bbb R$. – Asaf Karagila Dec 23 '24 at 12:09
  • THIS and THIS should help – Mark Viola Dec 30 '24 at 16:32

1 Answers1

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Distributions are not functions, at least not with domain and codomain $\mathbb{R}$.

For instance, consider the Dirac delta $\delta(x)$ and its distributional derivative. Both are zero except at $x = 0$, but they are very different.

Distributions are a generalization of functions in the following sense. For any continuous function $f:\mathbb{R}\to\mathbb{R}$, there is a continuous functional $\phi \mapsto \int f(x)\phi(x)dx$ on the space $C_c^\infty(\mathbb{R})$ of smooth functions of compact support.

And so the continuous dual space of $C_c^\infty(\mathbb{R})$ can be viewed as an extension of the space of continuous functions, equipped with such useful operations as addition, scalar multiplication, derivatives, and multiplication by certain functions.

mathperson314
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    I would qualify this: a distribution is a function, just not with domain $\mathbb{R}$. A distribution is a function whose domain - if memory serves - is the set of smooth compactly supported functions, and which is required to satisfy some basic properties. – Noah Schweber Dec 22 '24 at 22:10
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    @NoahSchweber Your comment is what I was looking for. I don't particularly care that the domain isn't $\mathbb R$, I just want to know that it is a function on something, much like a measure operates on collections of outcomes moreso than just individual outcomes. – Nate Dec 22 '24 at 22:16
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    I'd recommend one small change in your answer: Replace "$\varphi(x) \mapsto \ldots$" with "$\varphi \mapsto \ldots$", because $\varphi$ is the function itself, and $\varphi(x)$ is its value at a particular $x$ (which is meaningless on the left-hand side, and different from the $x$ on the right-hand side!). Every high-school kid says "the function $f(x)$", and so do some mathematicians...but by the time you're doing analysis, you end up with functions that may take other functions as arguments. If you named the distribution $Q$, would you talk about $Q(f(x))$ or $Q(f)$? I'd use the latter. – John Hughes Dec 22 '24 at 22:28