Talk:Semi-continuity

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[edit] This definition covers only R and not R^n

This definition only covers functions with f : X \rightarrow \mathbb{R}. The definition for f : X \rightarrow \mathbb{R}^n is important for fixed point theorems. --Clausen 23:31, 26 May 2005 (UTC)

I would challenge you to add a section at the bottom talking about R^n. :) This might be a better option than rewriting all the article in the general R^n case. Will you take the challenge? :) Oleg Alexandrov 00:25, 27 May 2005 (UTC)

[edit] Lower Semi-continous Example

I couldn't make any sense of this:

Imagine that you are scanning a certain scenery with your eyes and record the distance to the viewed      
object at all times. This yields a lower semi-continuous function which in general is not upper 
semi-continuous (for instance if you focus on the edge of a table).

Would someone like to clarify it? Or else we could just use a conventional example like f(x)= \lceil x \rceil Deepak 16:17, 31 March 2006 (UTC)

  • Ok then, I'm going to make the change. Deepak 16:52, 20 April 2006 (UTC)

Are we sure about the lower semi-continuous definition in terms of having a neighborhood around x_0 where all the x in the nbhd have f(x) > f(x_0) ? given that we are also saying that a continuous function is both upper and lower semi-continuous, then taking the identity function and fixing any point should give us a l.s.c. function where any neighborhood will have some x with f(x1) < f(x0) < f(x2) where nbhd is fixed around x0, x1, x2 are in the nbhd. Maybe just use the lim inf definition.

18:12, 8 October 2006 (UTC)chuck

I think the article is correct as it stands: the neighbourhood definition does not state that f(x) > f(x_0) in the neighbourhood, but rather states that f(x) > f(x_0) - ε, and this would fit OK with continuous functions such as the identity function. Madmath789 21:50, 8 October 2006 (UTC)
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