# Prove that $\int\limits_0^\pi {\frac{{\sin \left( {xt} \right)}} {t} \mathrm dt}$ is continuous

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!

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.