Sign up ×
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.

How can i prove that this function is continuous? $$ f\left( x \right) = \int\limits_0^\pi {\frac{{\sin \left( {xt} \right)}} {t} \mathrm dt} $$ Some hint? Don´t consider the zero in the endpoint of the integration zone, just take it as a limit $$ f\left( x \right) = \mathop {\lim }\limits_{\varepsilon ^ + \to 0} \int\limits_\varepsilon ^\pi {\frac{{\sin \left( {xt} \right)}} {t} \mathrm dt} $$ How can I do it? DX!

share|cite|improve this question
Is "DX!" some sort of hint? – Henning Makholm Oct 16 '11 at 17:42

1 Answer 1

up vote 1 down vote accepted

First of all, observe that $$ \lim_{t\to0}\frac{\sin(x\,t)}{t}=x\ , $$ so that the integral exists as a bona fide Riemann integral. Next, given $x,y\in\mathbb{R}$, $$ |f(x)-f(y)|\le\int_0^{\pi}\frac{|\sin(x\,t)-\sin(y\,t)|}{t}\,dt. $$ Now use the inequality $|\sin a-\sin b|\le\dots$ to conclude that $f$ is continuous.

share|cite|improve this answer
Thanks! I did it , it was easy D:!! – August Oct 16 '11 at 17:31

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.