Distributions are functions

>muh semantics
Deal with the fact that mathematicians are humans to and usually rely on linguistic shortcuts and context.
A "function" in the context of distribution theory usually means "locally integrable function on R^n", hence distributions are not "functions" in that sense.
Of course, distributions are functionals, hence mappings, but it would be misleading to call them "functions", that's all.

I bet you think the Dirac delta "function" is a function too, brainlets.

Does a function have to be defined on R? Functionals are just "functions" of functions no? Does this not count?

distributions is a word and not a functions

>mfw math undergrads don't know the set theoretical definition of a function

>using set theory
undergrad spotted

lol wat? what else should they be?

trees are plants. chicken are birds. op is a faggot. Shocking, I know...

How about citing a book?

>implying implications
Just because I know a basic definition doesn't mean I use it.

>but it would be misleading to call them "functions", that's all.
No it wouldn't.
Are you a retarded undergrad that thinks that the term function is reserved for [math]\mathbb{R}[/math] to [math]\mathbb{R}[/math]?

I clicked because I saw tits and I'm not ashamed to admit it!

>How about citing a book?
Take your pick.

>No it wouldn't.
Yes it would.

>Does a function have to be defined on R?
Only in what Americans call Calculus.
The rest of the world studies Analysis which implies basic knowledge of naive set theory.

But every density function is taking in something from IR and gives out something from IR (sometimes it's subsets of IR). What is the point?

#1 "in the ordinary sense"
Means that they are not real functions of real variables like they are known to the American undergraduate.
#2 "distributions are not functions OF TIME"
#3 Same as #1, see the attachment.
#4 Same as #1 and #3.
They only make it about semantics.

>Are you a retarded undergrad that thinks that the term function is reserved for R to R?
Can you read ?

No problem there, but you should be informed that from a strictly point of view that is actually a man.

Paul Dirac called it a function -- so you claim otherwise? L0Lno fgt pls

He is looking great these days

>He
What are you talking about?

What is the definition of a function? How does it apply or not apply to a distribution?

Not trying to be smug here, generally interested in the answer from your pov.

Being an asshole because pic related is trans

What is this scribbly D?

Does that mean amerimutts are brainlets?

What's wrong with set theory?

Yes

It not only claims that distributions are functions, but goes further to claim that functions are sets, which we pretend to be collections, but manipulate them as if they were extensional well-founded graph-theoretic trees, while using them to model logical properties.

tl;dr it needs strong typing

What is wrong with functions being sets? And type theory sounds like some CS monkey shit

>What is wrong with functions being sets?
On its own, nothing.
The problem is that once literally everything is a set, the property of sethood loses all meaning.

>And type theory sounds like some CS monkey shit
It is.
But it's also useful, and for some people that's pretty important. If you're not one of these people, then feel free to ignore it: you won't be missing out.

No. Set theory doesn't claim functions are sets.

The most basic definition for a function is that it is a RELATION. A bijection f:A->B.

Since a distrubution can give the same output for multiple inputs, it is not 1:1 between the sets. Therefore, it is not bijective. Therefore it is not a function. This is super basic shit, and there's literally no debate beyond semantics.

>The most basic definition for a function is that it is a RELATION. A bijection f:A->B.
Might want to reread your "Discrete Math for Ducks" textbook, freshman.

t. cs retard

>Every function is a bijection.
Why should we care about "sethood" being meaningful? Such a desire seems misguided.

>Why should we care about "sethood" being meaningful?
Because it's all you have.
The ontology of set theory consists of nothing but a single type -- the type of sets -- and it is in this sense that it carries no meaning.

Contrast this with a taxonomy like pic related for example, which classifies objects into multiple types based on the properties they have. Surely this is more informative, than the reductionist "everything is a set", and even if you really wanted that position, you can recover it by taking the transitive closure.

Its was merely a way to let everyone know. She is beautiful

The confusion about this distinction only arises because the term used to be reserved for maps from R to C or R, and distributions are functionals, which means that they map frunctions from R to C or R to R or C;

the dirac- delta, for example used to be defined as the function delta: R->R such that the integral over R of delta is one and int (f*delta) over R is f(0) for a certain kind of function (the L2-scalar product, in essence). However, it can be shown relatively easily that no function from R to R can satisfy these properties, and thus that the dirac delta has to be definded as a linear functional, at which point the integral notation becomes only that, notation.

Is there a reason beyond aesthetic (and perhaps ease) that you dislike defining those objects using sets? Not that that's a bad thing; I just still don't see why I should hope sethood is meaningful beyond it meaning that it's an object ZFC applies to.

>No. Set theory doesn't claim functions are sets.
Kek.

>No. Set theory doesn't claim functions are sets.
>The most basic definition for a function is that it is a RELATION
And relation is also a set. Relation between X and Y is just a subset of X×Y
>Since a distrubution can give the same output for multiple inputs, it is not 1:1 between the sets. Therefore, it is not bijective. Therefore it is not a function. This is super basic shit, and there's literally no debate beyond semantics.
Since when a function can only be a bijection?

I did not say that distributions were not functions. They are. But *idiomatically*, when people say "function", they mean "locally integrable function from R^n to R or C", which they identify as a (proper) subset of distributions, in which case some distributions are not "functions" *in that sense*.

There exist distributions that are neither continuous nor discrete. How is that a function?

They are defined as functions, but not from R to R. They're continuous linear functionals on the space of test functions, with respect to a given topology.

>How is that a function?
>I don't know what a function is.

>can't even answer a question
>thinks he can just call any mapping a function

brainlet