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?