If $f$ is continuous on $\Bbb R$ and uniformly continuous on every $(a,b)$ so it also uniformly continuous on $\Bbb R$

Question

*If there exists the same question with answer anywhere, plz delete this one, i didn't find one.

If $f$ is continuous on $\Bbb R$ and uniformly continuous on every $(a,b)$ so it is also uniformly continuous on $\Bbb R$

I think the statement is correct.

---

My attempt:

I think it is not true, since it can be that in every interval $(a,b)$ i can have a different gradient, which is constant for every interval, therefore, every interval has a finite $\delta > 0$ but with infinity number of intervals, each one may have a bigger gradient, in this case, i wont find $\delta > 0$ which is not dependent on $x$.

Therefore, the statment is not correct for every $f$.

Do i correct?

Thank you.

Answer 1

The statement is false.

Consider $f(x) = x^2$. Observe $f$ is continuous on $\mathbb{R}$ and \begin{align} |x^2-y^2|\leq |x+y||x-y|\leq 2\max(|a|, |b|)|x-y| \end{align} for all $x, y \in (a, b)$, i.e. $f$ is uniformly continuous on $(a, b)$. But is clear that $f$ is not uniformly continuous on $\mathbb{R}$.

Answer 2

The logic about what can go wrong is correct but this doesn't really prove the statement is wrong. You need a specific counterexample. Well, it is easy, take $f(x)=x^2$. Actually, any function which is continuous on $\mathbb{R}$ but not uniformly continuous there is a counterexample.