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)