Uniformly convergent & point convergent

Question

Let ${({f_n})_n}$ is a sequence that ${f_n}(x)=tan^{-1}(nx), x\in [0,\infty)$.

Prove for every $[a,b]$ that $a>0$ is Uniformly convergent and on $[0,b]$ just point wise convergent.

Answer 1

The sequence $(f_n)$ is point-wise convergent to the function $f$ defined by

$$f(0)=0\quad;\quad f(x)=\frac\pi2,\;\forall x>0$$ and since $f$ isn't continuous whereas the functions $f_n$ are continuous so the convergence isn't uniform on $[0,\infty)$