Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

My difficulty is the title problem. In the problem that is asked, I am attempting to show that there exists a positive continuous function $f$ on $\mathbb{R}$ so that $f$ is Lebesgue integrable on $\mathbb{R}$, but yet $\limsup_{x\rightarrow\infty} f(x) = \infty$.

The hint (title) tells me how I should construct my function.

I'm stuck because I'm not sure how to make $f$ continuous. I calculated my segment for values of $n=1,2,3$ and know that:

$f(1) = 1$ on $[1,2)$, $f(2) = 2$ on $[2, 17/8)$, and $f(3) = 3$ on $[3,82/27)$.

It's clear that the function's gaps get larger and larger as $n$ grows, and I'm not sure how to remedy this to make $f$ continuous. Hints/ideas would be greatly appreciated!

Thanks, Dom

share|improve this question

1 Answer 1

You have a problem at $2$ which can be rectified-just lower the upper limit of where the function is $1$ a bit. The basic idea is to take the function down to zero fast enough outside the intervals of the hint. So you could say $$f(x) = \begin {cases} 2 & 2 \le x \lt \frac {17}8 \\ 2-100(x-\frac {17}8) & \frac {17}8 \le x \lt \frac {17}8+\frac 1{50} \\0 & \frac {17}8+\frac 1{50} \le x \lt 3-\frac 1{100} \\100(x-(3-\frac 1{100}))&3-\frac 1{100} \le x \lt 3\end {cases}$$ Intuitively, each hump only contributes an area of $\frac 1{n^2}$ and we know the sum of those is finite. You need to specify a rule so the area contributed by the transitions is also finite. I just used $\frac 1{50}$ and $\frac 1{100}$ as examples.

share|improve this answer
    
That doesn't quite meet the desired specifications. The OP is looking for a positive function. –  Cameron Buie Nov 13 '12 at 1:40
    
@CameronBuie: good point-I missed that. But the section at $0$ can be shifted up by a proper $\epsilon$ to repair it. Again, we just need $\epsilon$ to fall fast enough that the flats contribute finite area. –  Ross Millikan Nov 13 '12 at 1:46

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.