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}
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!