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We play a game.

You give me any positive number, namely $\epsilon$ and any point $P$.

If I am able to find an $\phi$ such that $\forall x', x'' \in (P - \phi, P + \phi): |x'' - x'| \lt \epsilon$, then and only then I win.

If I won, the function is continuous at the point $P$.

The fact that given $f(x)$ is continuous at the $P$ does not guarantee continuity for not-$P$ points. As such, to prove continuity somewhere else, the whole game should be repeated?

But there is a different game - the one, which does not depend on particular point.

The only thing you provide is any positive $\epsilon$.

My goal is to give exactly the same response, with only difference: it has to be valid everywhere, i.e. for each and every point.

Winning second game means that the $f(x)$ is uniformly continuous.


Assuming that I haven't mistaken nothing, let me ask questions:

1) Continuity is and always be a local property?

2) Uniform continuity is and always be global property?

3) Is there a well-established definition of what "global" is in the given context?

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  1. Yes. A function $f$ is continuous if and only if, for each point $p$ of its domain, the restriction of $f$ to some neighborhood of $p$ is continuous.
  2. Yes.
  3. Yes. It is a global property because the fact that for each point $p$ of its domain, the restriction of $f$ to some neighborhood of $p$ is uniformly continuous is not sufficient to ensure that $f$ is uniformly continuus.
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