An example for integrable function that is never zero

Question

Let $f\::\mathbb{R}\to\mathbb{R}$ be integrable function for all $[a,b],\hspace{0.2cm} (a<b, \hspace{0.5cm} a,b\in\mathbb{R}$).

and$\hspace{0.2cm}\forall c,x\in\mathbb{R} \hspace{0.2cm} f(x)\not=0 \hspace{0.2cm} and \hspace{0.2cm}{\displaystyle \int_{c}^{c+1}f(x)\,dx}=0$

Can I have an example for such function?

Thank you!

$f(x)={1 if x\in[\frac {2k} 2, \frac {2k+1} 2]; -1 if x\in[\frac {2k+1} 2, \frac {2k+2} 2]}$ — user12986714, May 06 '20 at 19:15

Noob mathematician · Answer 1 · 2020-05-06T19:34:52.557

3

You see according to what you say let us take

$f(x)=\sin 2\pi x, \text{ for } x\in\mathbb{R- Z}$

$f(x)=1, \text{ for } x\in\mathbb{ Z}$

$$\int_{a}^{b}\sin 2\pi x \mathrm dx <\infty, \forall a<b$$

And $$\int_{c}^{c+1}\sin 2\pi x \mathrm dx=\int_{0}^{1}\sin 2\pi x \mathrm dx=0, \forall c$$.

edited May 06 '20 at 19:34

answered May 06 '20 at 19:26

1

The question also asked for $\forall x \in \mathbb{R}, f(x) \neq 0$, but $\sin(2\pi 0) = \sin(0) = 0$. – Chris Eagle May 06 '20 at 19:27
your answer works you just need to redefine sin where it is 0, to be something else, and it won't change the integral since it is a countable amount of points – BinyaminR May 06 '20 at 19:28
depends which kind of "integrable" he means but yeah – Peter Franek May 06 '20 at 19:34
@ChrisEagle I missed it sorry. – Noob mathematician May 06 '20 at 19:37
@BinyaminR Thank you – Noob mathematician May 06 '20 at 19:37
Thanks! I have one problem with your answer: if c=pi - is it still correct? I don't quite get it geometrically. – Ludwig May 06 '20 at 20:41
Yes it is because you see the function is almost periodic . All I mean changing the values at the isolated point won't effect this integral . This function is periodic. I don't know if you have studied measure theory, there is a notion called "measure zero set"( roughly speaking these are the sets of length zero). – Noob mathematician May 06 '20 at 20:49
Periodicity ensures the second property . – Noob mathematician May 06 '20 at 20:50
@Ludwig Have a look at this as well – Noob mathematician May 06 '20 at 20:54
@Ludwig is the answer not worthy to be accepted? – Noob mathematician May 19 '20 at 15:19

1 Answers1