let $f$ be a continuously differentiable function on $\mathbb R$.
let $f_n(x)$= $n(f(x+\frac{1}{n})-f(x))$
then
1) $f_n$ converges uniformly on $\mathbb R$
2) $f_n$ converges on $\mathbb R$ but not necessarily uniformly
3) $f_n$ converges to derivative of $f$ uniformly on $[0,1]$
4) there is no guarantee that $f_n$ converges on any open interval
i know that result for any interval so from that i know $(3)$ holds. but how to see for $\mathbb R$.
$(2)$ is also answer
Hint: $$n(f(x+1/n)−f(x)) = \frac{f(x+1/n)−f(x)}{1/n} \to f'(x)$$ (at least) pointwise in $\Bbb R$. And for (1) try with some easy examples.