calculus - Prove that $intlimits_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!

Answer

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.