Tuesday, 29 January 2019

calculus - Prove that intlimitsp0ifracsinleft(xtright)tmathrmdt is continuous



How can I prove that this function is continuous? f(x)=π0sin(xt)tdt
Some hint?
Don´t consider the zero in the endpoint of the integration zone, just take it as a limit f(x)=lim
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.


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