Uniform Continuity of a Sequence of Functions with a Piece-Wise Defined Limit Consider the sequence of functions
$$h_n(x)=\frac{x}{1+x^n}$$
over the domain $[0,\infty)$.
I found its pointwise limit to be
$$h(x)=\begin{cases}
x & \text{ if } 0\leq x<1\text{,}  \\ 
1/2 & \text{ if } x=1\text{,}  \\ 
0 & \text{ if } x>1\text{.} 
\end{cases}$$
But I have a "gut-feeling" that the function may not be uniformly continuous over that interval (because it is piece-wise defined), so I am now trying to find a smaller set for which it is:
I naturally went with the choice of $(0,1)$, but then
$$\left|h_n(x)-h(x)\right|=\left|\frac{x}{1+x^n}-x\right|=\frac{x^{n+1}}{1+x^n}<x<\epsilon,$$
which does not provide the necessary information I need to conclude that the function is uniformly continuous over that interval.
Do you guys have any ideas? Thanks!
 A: Hints:
As noted in the comments,  $(h_n)$ does not converge uniformly on $[1,\infty)$ or on $[0,1]$, since its limit function is discontinuous on those intervals.
By considering the values $x_n=1-{1\over n}$, you should be able to show that $(h_n)$ does not converge uniformly on the interval $[0,1)$ (compute $\lim\limits_{n\rightarrow\infty} h_n(x_n)$). By considering $x_n=1+{1\over n}$, you should be able to show that $(h_n)$ does not converge uniformly on the interval $(1,\infty)$ (note this would also verify the results of the preceeding paragraph).
But, consider what type of convergence you have on a set of the form $[0,1-\epsilon]$ or $[1+\epsilon,\infty)$ for a fixed $\epsilon>0$. Here, use the inequalities
$$
\Bigl| {x\over 1+x^n} -x  \Bigr|=\Bigl| {x^{n+1}\over 1+x^n}\Bigr|\le x^{n+1},
$$
for the $[0,1-\epsilon]$ case; and
$$
\Bigl| {x\over 1+x^n}   \Bigr|\le\Bigl| {x \over  x^n}\Bigr|={1\over x^{n-1}},
$$
for the $[1+\epsilon,\infty)$ case.


Looking at the  graphs of the $h_n$ should help:

 
Above are plotted $\color{darkred}{h_2}$, $\color{darkslateblue}{h_4}$, $\color{olivedrab}{h_{10}}$, $\color{salmon}{h_{40}}$, $\color{cyan}{h_{60}}$, $\color{yellow}{h_{100}}$,
and  $\color{pink}{h_{1000}}$ (the yellow is $h_{100}$; it's barely readable on my monitor).
