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(Context here is I’m thinking about functions and these questions were part of some used in the 80s to explore understanding of functions. I wonder what’s changed in the last 40 years?!)

@LearningMaths @LearningMaths "function" is a very generic term, anything that associates one value of the codomain for every value of the domain is a function (with domain and codomain being arbitrary sets)

The property you are looking for, based on your question, is that of a continuous function.

@sophieschmieg I’ve lifted the questions directly from this paper

Vinner, S., 1983. Concept definition, concept image and the notion of function.

Which explores what high school students understand a function to be.
The distribution of responses here vs those on Twitter is quite different!

@LearningMaths interesting. I did learn the definition of function as a generic mapping in high school, but it might be different in different countries

@sophieschmieg the article is a study carried out in Israel. I’m UK based. It’s interesting to hear what people’s understanding is.

@LearningMaths that's fascinating

@rrogers What do you think a "Generalised Function" is?

What is a "Non-Generalised Function"?

That's the point of this poll/question ... to understand what people thing "function" means. It has a specific meaning in maths that not everyone, and especially not every student, has assimilated.

CC: @LearningMaths

@ColinTheMathmo @LearningMaths
oops:"Generalised->"Generalized
The Generalized

"Generalized Functions" and ilk
basically result from allowing an integral limit process.
The prototypes came from Heavyside and were embedded in QM by the "Dirac Function" The mathematical/axiomatic exposition was basically done by Abraham Robinson as "Non-Standard analysis".
In addition, it's tacitly used in Laplace/Fourier Transforms, but that still carries an implicit integration.
https://en.wikipedia.org/wiki/Dirac_delta_function
https://en.wikipedia.org/wiki/Nonstandard_analysis
https://en.wikipedia.org/wiki/Oliver_Heaviside
https://en.wikipedia.org/wiki/Generalized_function

I can write out the limit process but it's on Dirac's wiki page and in fact, there are a whole set of "test functions" that work as well.
DLMF calls them distributions
probably because they are formally related to PDFs in probability theory.
https://dlmf.nist.gov/1.16
A fairly standard reference book is:
Function and Generalized Function Transforms
By Zayed.

@rrogers So I am tempted to conclude that you think "function" means more than simply deciding what f(x) is for each x in the domain. You seem to require that a "function" has other characteristics.

If so ... what?

CC: @LearningMaths

@ColinTheMathmo @LearningMaths
Yes I realize from the basics that
y=f(x) can be defined by a selection of a particular set of
(x,y) pairs and then calling the selection a function.
and y has to "exist" for all x.
But I normally don't go that far into the weeds.
I guess I should apologize.
In Real Analysis, I did have to deal with Cantor's functions and such. (personally "the march of the Monster functions")

@rrogers I think part of what @LearningMaths is getting at is to discover what additional features people assume a function will have.

Students coming in to 1st year university maths often seem to think that a function must be specifiable by a single equation or expression, or at most two. They also seem to think that all functions must be continuous expect perhaps at one or two points, and differentiable, except perhaps at one or two points.

They then get very confused when the lecturer or tutor will say "function" and simply mean:

"Blob A, Blob B, Arrow from A to B"

... and not assume any characteristics.

That's why I'm interested to see what you think "function" implies.

CC: @homomorphism@tech.lgbt

@ColinTheMathmo @rrogers @LearningMaths @homomorphism@tech.lgbt Right, but typical 1st year students have never heard of Heaviside nor Dirac Delta, have varying degrees of familiarity with continuity and differentiability and smoothness and such.

Absolute precision is less and less achievable the lower the sophistication of the audience, and terminology is all about effective communication.

Since the poll was addressed to "maths teachers", I assume it's not assuming a high level of sophistication, but more about communicating with students.

@dougmerritt Agreed to everything. It's interesting to see what students think functions are, and then help them (the students) see them (the functions) as rather more general and less limited. Then their (the students') friends (the "nice" functions) can be seen as common special cases.

One place where communication regularly fails is that mathematicians (and teachers) will assume certain domains and assume certain properties as being implied by an unspoken context.

Therein lie potential problems, and making this all explicit early can be really useful.

CC: @rrogers @LearningMaths @homomorphism@tech.lgbt

Charting this ... I think it's really interesting:

Calling @Chartodon ...

#MTBoS #MathsEd

@karenshancock @dougmerritt @rrogers @LearningMaths @homomorphism@tech.lgbt

@ColinTheMathmo

Your chart is ready, and can be found here:

https://www.solipsys.co.uk/Chartodon/110810778219358428.svg

Things may have changed since I started compiling that, and some things may have been inaccessible.

In particular, the very nature of the fediverse means some toots may never have made it to my instance, in which case I can't see them, and can't include them.

The chart will eventually be deleted, so if you'd like to keep it, make sure you download a copy.

@dougmerritt I find it *much* easier to follow who is replying to whom, and I frequently find posts I missed. I especially get to see the context for some posts which can otherwise be hard to find.

In this case it's all been pretty linear, but sometimes discussions are a lot more sprawling, and then I find the chart indispensable.

@karenshancock @rrogers @LearningMaths @homomorphism@tech.lgbt

@ColinTheMathmo @karenshancock @rrogers @LearningMaths @homomorphism@tech.lgbt Ahh...I could see that; thanks.

@dougmerritt Also, sometimes it's just so I can refer to the content later.

Here's an example of that:

https://www.solipsys.co.uk/Chartodon/110808068619687656.svg

And here's an example of a larger discussion:

https://www.solipsys.co.uk/Chartodon/110768815698345202a.svg

@karenshancock @rrogers @LearningMaths @homomorphism@tech.lgbt

@ColinTheMathmo @karenshancock @rrogers @LearningMaths @homomorphism@tech.lgbt And it's an instance of the utlimate, the Galactic Modeling Language! :)
https://kidneybone.com/c2/wiki/GalacticModelingLanguage

(I guess I'll have to look into why the search hit was on kidneybone.com...)

But yeah, all that does seem handy.

@dougmerritt

Re the GML: Indeed

(I'm rather pleased I have a near complete copy of the C2 wiki content. I might do something with it one day.)

@karenshancock @rrogers @LearningMaths @homomorphism@tech.lgbt

@rrogers I, on the other hand, am perfectly happy with this:

Let f:R->R be such that:

* When x is irrational, f(x)=0;
* When x =a/b in lowest terms, f(x)=1/b.

I also work in Graph Theory, and saying "Let G and H be graphs, and let f be a function from V(G) to V(H) ..."

I don't assume anything about f except that every vx of G gets carried to a vx of H.

CC: @LearningMaths @homomorphism@tech.lgbt

@ColinTheMathmo @LearningMaths @homomorphism@tech.lgbt
And I have always thought that, perhaps, it should be called something else :) :)
To most people function would imply the domain of Z, R, C, and derivatives (not mathematical) thereof.
I remember my first introduction to the (y,x) idea in some library book in H.S., and thought that's kind of neat, but seems a little pedantic. Doesn't seem to be relevant to anybody that isn't a college senior in math.

@LambdaDuck would ‘assigns’ work better (suggested by @ColinTheMathmo)?
This set of questions is a translation from (I think) Hebrew - I’ve used the exact wording from an article published in 1983.
I’m interested in peoples interpretations and understanding so read it as you like!