Talk:Continuous function (topology)

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[edit] Balance of formality and accessibility

I have rewritten the introduction (again) in a way which is appropriate for an article on topology, but emphasising that the definition for metric spaces is exactly the same as the topological one. Using neighbourhoods rather than open sets is far clearer, because continuity is essentially about what happens near individual points. The reason the very abstract open set formalisation works is that an open set is simply any set that contains a neighbourhood of every one of its points, so continuity in terms of open sets is about continuity in parallel at every one of the points in the sets.

The reason I object to a too informal introduction is that it is easy to give a false idea of what continuity is. It is easy to define a function f on the plane which is discontinuous, but for which limits of sequences are preserved (i.e. any sequence {x_n}with a limit L is mapped to {f(x_n)} with a limit f(L). Anyone reading an informal definition about nearness could easily assume it would be enough for limits of sequences to be preserved.

There is still some stuff late in this article which I think is misleading because it ignores the fact that functions don't have to be injective. Elroch 01:05, 27 April 2006 (UTC)

Are you sure about this claim that a discontinuous function can still preserve limits of sequences? I am under the impression that in any first countable space, a function is continuous if and only if it preserves limits of sequences. In fact, I added material asserting that fact to the article a few weeks ago. -lethe ^{talk +} 01:29, 27 April 2006 (UTC)

You are right. I was thinking of the function:

and hastily assumed it preserved limits of sequences at the origin. But it doesn't, and your general result is one I recall to be true. However my observation is true for spaces that are not first countable, which is relevant for a general topological concept. Elroch 11:52, 27 April 2006 (UTC)

Yes, indeed, your observation is still true for non-first countable spaces. Then you have the slightly generalized version which says that a function is continuous if and only if it preserves limits of nets (generalized sequences). Many people define continuity this way, in fact, so this view should be represented in the article. -lethe ^{talk +} 20:20, 27 April 2006 (UTC)

I favour sacrificing formality for accessibility especially in the intro, but I think yours is better than the "nearness is measured..." wording for both qualities. Thanks.

As for misleading stuff later in the article, if you are referring to the statement that any neighbourhood V of f(x) contains an image f(U) of a neighbourhood U of x, rather than the statement that the inverse image of a neighourhood of f(x) contains a neighbourhood of x: my understanding is that the former is also correct even when f is not injective. Am I mistaken? -Dan 14:18, 27 April 2006 (UTC)

The thing I was referring to was about the topology induced by a continuous function. I elaborated this by explaining the quotient topology and dealing with functions that are not necessarily surjective. Elroch 15:03, 27 April 2006 (UTC)

The stuff about the quotient topology is a bit misleading. Every function from a set determines a final topology. Every function also determines an equivalence relation (x~y iff f(x)=f(y)). If the function is surjective, it's true that the quotient space and the final topology on the codomain are homeomorphic. I wouldn't say that they're equal though; the quotient space is a space of equivalence classes, while a general codomain need not be. I've cleaned up the section, and also included the dual case, but now we've got two long paragraphs of stuff that I'm not sure belongs in this article. -lethe ^{talk +} 21:34, 27 April 2006 (UTC)