1
$\begingroup$

Today, my teacher asked about a real function that was integrable, but such that $\displaystyle\lim_{t\to \infty}f(t)\neq 0$ ($f$ need not have a limit).

It's easy to forge an example: a function that has tightening spikes each with areas $1/n^2$ does the trick.

He then asked for a real $C^\infty$ function that was integrable, but such that $\displaystyle\lim_{t\to \infty}f(t)\neq 0$ ($f$ need not have a limit).

One heuristic argument is to take the previous function and "smoothen" the spikes. However, he referred to a nicer function that can be defined explicitely.

Do you have an idea ? I've looked for something involving $\sin,\cos,\tan$

$\endgroup$
  • $\begingroup$ Are you familiar with standard bump functions? See en.wikipedia.org/wiki/Bump_function $\endgroup$ – guest196883 Jun 10 '14 at 18:33
  • $\begingroup$ @SamDeHority Yes, what do you plan on doing with them ? $\endgroup$ – Gabriel Romon Jun 10 '14 at 18:34
  • $\begingroup$ Does "integrable" refer to Riemann or Lebesgue? $\endgroup$ – Daniel Fischer Jun 10 '14 at 18:41
  • $\begingroup$ @DanielFischer Riemann I guess ($\int_0^\infty f(t)$ converges) $\endgroup$ – Gabriel Romon Jun 10 '14 at 18:42
  • $\begingroup$ In that case, cnick's answer gives an explicit example. $\endgroup$ – Daniel Fischer Jun 10 '14 at 18:43
2
$\begingroup$

why not something like $f(x) = \sin(x^2)$? It has to oscillate faster as you go towards infinity so that it adds/subtracts less and less area each time in order for the integral to converge.

$\endgroup$
1
$\begingroup$

HINT: Try to construct "smooth mountains" by using the Cauchy function

$$h:x\mapsto \begin{cases} e^{-\frac{1}{x}} &\text{if }x>0\\ 0&\text{if }x\leq 0 \end{cases}$$

which is very useful for constructing $C^\infty$ functions with that kind of properties.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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