What is the difference between a function defined locally at $0$ and globally at $0$ on a set $S$? My textbook keeps referring to these things, but I couldn't find any definition about it anywhere. Can someone please elaborate?
Thanks in advance!
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3If $S$ has a topology, e.g. is the real line, then being defined locally means being defined on some neighborhood $U$ of zero, and being defined globally just means being defined on the whole set. But "defined globally at zero" is a strange thing to say. If you can post a more exact quote exemplifying your textbook's usage, we can probably answer better. – Kevin Carlson Feb 07 '15 at 20:01
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@KevinCarlson Thanks for your answer. I am reading Spivak's book Calculus on manifolds. Here is the question I took it from link I am studying partition of unity, see Ted Shifrin's answer please – Marion Crane Feb 07 '15 at 22:27
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In the general context of topology, the term “locally at $x ∈ S$” most commonly means “for a local basis of $x ∈ S$” – This of course begs the question: What is a local basis?, which will take you to the even more fundamental questions:
- What is a neighbourhood?, and
- What is a topological space?
If you don’t know these notions yet, you don’t have to learn them to understand what is meant in the context of defining functions in real analysis:
For $S ⊂ ℝ$, your textbook probably takes a function $f$ to be defined locally at $x ∈ S$ if there is a neighbourhood $V$ of $x$ on which $f$ is defined, – i.e. a set $V ⊂ S$ such that for some $ε > 0$, the $ε$-ball $B_ε(x) = \{y ∈ S;~|y - x| < ε\}$ is contained in $V$.
A function that is globally defined on $S$ should just be the same as a function on $S$. The term is probably only used for the contrast to something which is locally defined. So “globally defined” means “defined, … and not just locally!”.
See also this related question, concerning the different concept of local properties with which you shouldn’t confuse this, as explained in the answer by Najib.
