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Here's a proposition I find hard to understand in my professor's notes. With N(c,r) he regards the r neighborhood of point c. I couldn't quite figure out what he defined by that sigma notation. If anyone could please clarify that'd be great.

From what I understand so far, he claims that if we have a set of functions continuous on a interval, then if we define a function g that for each x in the interval returns the minimal value for x in {f1(x), ... fk(x)}, that function would be continuous as well. Did I get it right?

The proposition and its proof

Noa
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  • For each $x$ the notation $\sum(x) $ is a finite set of indices (integers between $1$ and $K$) and not a function. I agree the notation could have avoided the $() $ so that it does not resemble a function. – Paramanand Singh May 18 '17 at 07:23
  • So all this sigma means is all the points x where one of the functions fj equals the function g defined? – Noa May 18 '17 at 08:34
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    Well it means those indexes $j$ and not the $x$. – Paramanand Singh May 18 '17 at 08:41
  • That's what I thought. Thanks! How about the rest? is this entire thing equivalent to what they did here? https://math.stackexchange.com/questions/530135/prove-that-the-min-and-max-of-2-continuous-function-are-continuous – Noa May 18 '17 at 08:44
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    Yes it is equivalent, but the presentation in the linked question in your last comment is much better. You should use that for your understanding. – Paramanand Singh May 18 '17 at 08:46
  • I think so too, the more I look into my professor's proof it seems unnecessarily complicated. Thank you so much for you help! – Noa May 18 '17 at 08:48
  • The proof by your professor uses too many symbols and that is a big obstacle to understanding. Humans have evolved to understand languages of communication like English or German or Hindi etc much better than mathematical symbols. They should not be used unless necessary. – Paramanand Singh May 18 '17 at 08:50

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