What if the interval $[0,1]$ changes to $\mathbb R$ in the question?

Question

What if the interval $[0,1]$ changes to $\mathbb R$ in the question Let $f(x)=\sum\limits_{n=1}^\infty \frac{1}{n} \sin(\frac{x}{n})$. Where is $f$ defined? Is it continuous? Differentiable? Twice-Differentiable?? Will the uniform convergence and differentiability have any issue? I understood the proof given in the answer provided by Bernard. How do I extend the proof to $\mathbb R$?

Answer 1

You can extened these resulets on $\Bbb R$ as you can do it on each $[-M,M]$ for $M>0$ and continuity, differentiablity are local properties, means they are depend only on the behavior in some neighbourhood.

Let $x\in\Bbb R$, then there is $M>0$ such that, $|x|<M$. Now, $f\big|_{[-M,M]}$ is continuous and differentiable. Hence $f$ is continuous and differentiable on $[-M,M]$, in particular at $x$.