Definition talk:Continuous Mapping

Separate pages
I would propose to put the real, metric space and topological space definitions in different pages. Each of them can grow a lot, as this concept is very common and has a huge quantity of equivalent statements and related results. Also, each of them may need different explanations, and it would be easier to refer to a particular one if it is on a separate article than linking to the particular section.--Cañizo 00:53, 19 February 2009 (UTC)

Not happy about that. Although there are different explanations depending on the context, the concept as a whole is one that requires to all be kept together so as to provide the context for the other definitions. The plan is (eventually) to make the connection between all these definitions at the topological level (the most general), and to indicate that they all mean the same thing. There's more work to be done yet and it all needs to be kept in place or there's a danger of losing the thread of where this is going.--Matt Westwood 06:33, 19 February 2009 (UTC)

Ok, let's see how it goes. I still think it would be clearer if they are in separate pages, especially leaving one page for the definition on $$\R$$ with limits, and one for the topological definition, but we can wait and see.--Cañizo 12:04, 19 February 2009 (UTC)

Different definition
Another suggestion is that one could make the definition simpler by saying the following:

Let $$I$$ be a real interval, and let $$f:I \to \R$$ be a function.

The function $$f$$ is said to be continuous at a point $$x \in I$$ if
 * $$\lim_{\underset{y \in I}{y \to x}} f(y) = f(x).$$

Note that the limit is taken among points that belong to the interval.

The function $$f$$ is said to be continuous on $$I$$ if it is continuous at every point of the interval $$I$$.

Then one can give some examples to clarify that the limit is a limit from below or above at the endpoints of an interval. This way the definition is less complicated. Otherwise, it looks like there are lots of special cases to be taken into account.--Cañizo 00:53, 19 February 2009 (UTC)

There are lots of special cases to take into account. There's a danger of glossing over the fact that a function may have different limits at a particular point on the interval, which is why the care to define it on the left and the right. --Matt Westwood 06:29, 19 February 2009 (UTC)

But the concept of limit at a point on a set is the same, no matter where on the set the point is. When one defines a continuous function on a metric space with the analogous definition, nobody is going to separate every point on the boundary of a domain as a special case. I think it makes the definition longer and more complicated... If you think one may overlook the fact that limits at the boundary must be taken from inside, this could be clearly stated as a comment after the definition.--Cañizo 12:08, 19 February 2009 (UTC)

Shrug. See how you go at putting something together that looks okay and holds all the separate contexts together. Note that the intention was originally to base it on the definition of limit.

If you do decide to delete swathes of stuff, please put it between comment delimiters   instead of physically removing it. Or I'll hate you. ;-) --Matt Westwood 21:45, 19 February 2009 (UTC)

Ok, I'll try to reorganize a bit, let's see.--Cañizo 00:09, 21 February 2009 (UTC)

I did change "swathes of stuff", as you said... It still needs a lot of improvement. It is hard to decide how to put things for a basic definition like this! Where should the examples go? The notes? The explanations? Also, when one writes a formal definition or theorem, it is useful to separate it clearly from the surrounding text. Do you always use a separate heading? I tried writing Definition. at the beginning of the paragraph, but it doesn't look that good, and one cannot link later to the specific definition... I guess that using one section for one definition or theorem is ok, but I'd say it is important to make it clear that it is so. That's why I wrote "Definition:" in front of the section headings... At some point the site should have some kind of guidelines for this.--Cañizo 02:26, 21 February 2009 (UTC)

I personally think this version looks good. My only (major) concern is that we have the definition for topological spaces last, even though it is the most general. We may want to rethink the order of that. --Cynic (talk) 03:32, 21 February 2009 (UTC)

Good job
I agree - exactly the approach I was trying to aim for (although I'd be tempted to add a few internal links, I may get round to that later, I haven't got time today, I'm on catering duty today).

I'm not sure I believe the topological space definition should come first. My thinking is: as you "evolve" in your level of mathematical sophistication, first you meet the definition for real numbers, then you do the course on complex analysis and learn about that, then on the way towards topology you study metric spaces, and last of all at the pinnacle of your undergraduate career you are initiated into the arcane truths of topology. Facetiousness aside, I crafted this page (and to an extent certain other related pages and definitions) with that progression in mind. I wasn't consistent in this - other pages have the topological one first and the less general but more accessible cases following on as specific examples. Depending on the page, or the concept, one or the other approach may be better - but in this case I think the "evolution" approach may work best.

It would be a (worthily?) ambitious project to make all these definitions of continuity "holistic" to the extent that one definition naturally leads straight on to the next. It would be worth adding how it extends to complex numbers, and from there to the more general case of metric spaces. From there via the concept of an "open set" to the natural extension to topology.

At the moment there's a break in the flow at the point where metric spaces starts. This is because the latter definition has no reference to the concept of the "limit", whereas that for the real number space is (now) perfectly balanced. Ultimate would be to define metric spaces in exactly the same language and structure, and then the conceptual flow would be unimpeded by the response that it appears that the definitions are unrelated. Despite (or perhaps because of) my distaste for the tedium of the detail of real analysis, the bigger picture of continuity, convergence and closure becomes all the more exciting when it is revealed that all the fiddling around with details is no more than that - detail within a global generalised master plan of simplicity.

As for the details of your questions: adding "Definition" to the front of the definitions for stuff doesn't seem to do any harm to the appearance or readability of the page, so no problem there. Note that when you make it a section, you can then link directly to it by means of, for example: continuity from the left at a point. So you can link to the specific definition. Even in the same page, I found.

One small stylistic note: our "house style" has evolved so that the titles of things have the significant words capitalised. IMO this makes it look more "professional". When you link to such a title, you can always make the appearance text as lowercase as you like (and in that context seems to be appropriate).

Feel free to browse with "random proof", for example, to see what's generally evolved. --Matt Westwood 12:50, 21 February 2009 (UTC)