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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!

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    This is what I now have: Over the set $(1,\infty)$, we see that |h_n(x)-h(x)|=|x/(1+x^n)-0|=x/(1+x^n)<1/2. Is this sufficient to show that the function converges uniformly over that interval?2012-03-07

1 Answers 1

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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:

enter image description here

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).