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)=limε+0πεsin(xt)tdt

How can I do it? DX!


Answer



First of all, observe that
limt0sin(xt)t=x ,


so that the integral exists as a bona fide Riemann integral. Next, given x,yR,
|f(x)f(y)|π0|sin(xt)sin(yt)|tdt.

Now use the inequality |sinasinb| to conclude that f is continuous.


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