Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm looking for Riemann integrable function $f:\mathbb{R}\to\mathbb{R}$ with $\int^{a+1}_a f(x)dx=0$ for all $a\in\mathbb{R}$, but $ f(x)\neq 0$.

I suspect that floor function involves here, if so, then how?

Thank you all!

To clarify: $f$ must not be identically equal to $0$, and it should be integrable over any finite interval.

share|cite|improve this question
It would be good if you clarified what you mean by $f(x)\ne0$. Does that mean that $f$ can never take the value $0$? Or rather that $f$ should not be identically equal to $0$? As you see, some of the examples below may or may not work, depending on what you are asking. – Andrés E. Caicedo May 11 '13 at 16:46
I meant that $f$ should not be identically equal to $0$. – Sandra West May 11 '13 at 16:49
Then Lana's (user:77181) example works fine, and then the question becomes whether a less "silly" example is possible. Anyway, could you clarify: By "Riemann integrable", do you mean that the improper integral $\int_{-\infty}^\infty f(x)\,dx$ exists, that the improper integral of $|f|$ exists, or that the integrals over finite intervals exist? – Andrés E. Caicedo May 11 '13 at 16:54
The integrals over finite intervals exist. – Sandra West May 11 '13 at 16:57
Ah, ok, thank you for the reply. Then the other two examples show a general approach. I would suggest to edit the question so these clarifications are not buried in the comments. – Andrés E. Caicedo May 11 '13 at 16:59
up vote 9 down vote accepted

What about the characteristic function of a singleton?

share|cite|improve this answer
How I do that...? – Sandra West May 11 '13 at 17:16
What is there to do? – Andrés E. Caicedo May 11 '13 at 17:18
Ok..... got it! – Sandra West May 11 '13 at 17:29
what is characteristics function of singletons? explain for me please. I do not understand your one line answer. – Un Chien Andalou May 17 '13 at 8:14
Pick $x \in \mathbb{R}$, the characteristic function $ \chi: \mathbb{R} \to \mathbb{R}$ of its singleton $ \{ x \}$ it's defined as follows: $\chi(z):=1$ iff $z=x$, otherwise $\chi(z):=0$. – Edoardo Lanari May 17 '13 at 12:23

If you mean "Riemann integrable on every finite interval", try $f(x)=\sin{2\pi x}$. If it needs to be non-zero everywhere, you may redefine it to be $1$ for $2x\in\mathbb Z$.

share|cite|improve this answer
If you want it to be integrable as an improper riemann integral over the whole real axis, simply damp it with $e^{-x^2}$ – fgp May 11 '13 at 16:37
@fgp You wont then have $\int_{a}^{a+1}$ to be invariant for all $a$. – user17762 May 11 '13 at 16:39
Hm, true. I missed that. – fgp May 11 '13 at 16:47

Consider any integrable periodic function with period $1$, which integrates to $0$, i.e., let $g(x)$ be any function defined on interval $[0,1]$. Then consider $$f(x) = \begin{cases} g(x) - \underbrace{\int_0^1 g(x) dx}_b & \text{if }x\in[0,1]\\ g(\{ x\}) - b & \text{else}\end{cases}$$

share|cite|improve this answer
This is surely not improperly Riemann integrable over all of $\mathbb{R}$. – Martin May 11 '13 at 16:36
@Martin Oh ok. I interpreted the question as Riemann integrable over any unit interval, which I believe is what the OP is looking for. – user17762 May 11 '13 at 16:38

How about trigonometric functions?

share|cite|improve this answer
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – vadim123 May 11 '13 at 23:01
@vadim123, it sure does... think $\sin x$ between 0 and $2 \pi$. – vonbrand May 11 '13 at 23:18
@vonbrand, in my understanding of Math.SE policy short answers like this should be comments instead. I did not mean to comment on its relevance or usefulness, unfortunately the "low quality post" selection menu inserted this text automatically on my behalf. – vadim123 May 12 '13 at 0:12
@vadim123, while answers which elaborate are usually desired, in this case being brief is pretty reasonable! – Mariano Suárez-Alvarez May 12 '13 at 5:41
@vadim123: a short answer that has gotten a fair number of votes. – robjohn May 12 '13 at 9:58

Your Answer


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.