Let $f : \Bbb R \longrightarrow \Bbb R$ be an arbitrary everywhere continuous function. Let $I \subseteq \Bbb R$ be an interval. What can we say about the uniform continuity of $f$ on $I$ where $I$ is respectively open, closed and bounded?
For open and bounded cases I can restrict the everywhere continuous function $f(x) = x^2$ on $(0,1).$ Then $f$ is not uniformly continuous.
Now what can I do for closed intervals? Can I get the same kind of counter-examples? If that closed set is bounded then it is compact and we know that any continuous function on a compact set is uniformly continuous. So if we want to get a counter-example we need to work with closed and unbounded sets one of which is $\Bbb R$ itself and the same function $f(x)= x^2$ will work for this case also. So we are done with all the cases.
Is my reasoning correct at all? Someone please verify it.
Thank you very much.