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Give an example of a real function $f$ and a convergent sequence $\{x_n\}$ with limit $l$ for which $\{f(x_n)\}$ is convergent but its limit is not $f(l)$.

I know $f$ must be discontinuous but can't think of a simple example that makes sense. Any hints would be appreciated. Does anyone have any advice on how I should be approaching "give an example" problems in general?

Many thanks.

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2 Answers 2

up vote 3 down vote accepted

The general thing to keep in mind with these problems is to just build your example a step at a time, keeping everything as simple as possible.

First we need a convergent sequence $x_n$. The simplest possible convergent sequence is a constant sequence, but that can't work here. The next simplest is probably $x_n=1/n$, converging to $0$.

Now we need $f(x_n)$ to converge. Again, the simplest way for this to happen is for $f(x_n)$ to be constant, so let's set $f(1/n)=0$ for all $n$.

Finally, we need $f(l)$ to not be the limit of $f(x_n)$. In other words, $f(0)$ has to not equal $0$. This is easily accomplished, just set $f(0)=1$ say.

All that remains is to define $f$ for all other points. Since $f$ already satisfies all the conditions, we can do this however we like. For the sake of argument, we could set $f(x)=0$ wherever $f(x)$ wasn't already defined.

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Thanks a lot, your step-by-step was very helpful. I did end up with an answer in the end but through a much more convoluted method. Cheers! –  vim Jun 25 '13 at 10:31

To find an example where something breaks, take an example where it works and change the thing that makes it work.

Pick your favorite convergent sequence $\{x_n\}$ with limit $l$, and pick your favorite continuous function $g$. Then (as you know) $\{g(x_n)\}$ is a convergent sequence with limit $g(l)$.

Then define your desired function $f$ by $$f(x)=\begin{cases} g(x) & \text{ if }x\neq l,\\ \text{anything other than }g(l) & \text{ if }x=l. \end{cases}$$ (technically this only works if there isn't an $N$ such that $x_n=l$ for all $n\geq N$, but that'd be a pretty boring convergent sequence to have as your favorite).

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