I tried working with this sequence $\displaystyle f_n(x) = \frac{x^n}{1 + x^n}$ and I have found the following:
1) It is not uniformly convergent on $[0,\infty)$ as the limit function $f$ is not continuous. $f$ is $0$ in $[0,1)$, $1/2$ at $x=1$ and $1$ in $(1,\infty)$.
2) It is not uniformly continuous in $[0,1)$ because $f_n$ is increasing in $x$ and so $\sup\{|f_n(x)-f(x)|:x\in[0,1)\} $ does not tend to zero.
3) It is uniformly convergent in $(1,\infty)$ by a simple inequality that allows the supremum to tend to $0$. $$\frac{1}{1+x^n}<\frac{1}{x^n}.$$
Am I right in my observations?
1) and 2) are correct. For 3) note that $$\sup_{x\in (1,+\infty)}\left|1-\frac{x^n}{1 + x^n}\right|=\sup_{x\in (1,+\infty)}\frac{1}{1+x^n}=\frac{1}{2}.$$ and therefore $f_n$ is NOT uniformly convergent in $(1,+\infty)$. On the other hand, if $a>1$ then
$$\sup_{x\in [a,+\infty)}\left|1-\frac{x^n}{1 + x^n}\right|=\sup_{x\in [a,+\infty)}\frac{1}{1+x^n}=\frac{1}{1+a^n}\to 0.$$