Talk:Continuous function (topology)

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)[reply]

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)[reply]