How can I prove that this function is continuous? f(x)=π∫0sin(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+→0π∫εsin(xt)tdt
How can I do it? DX!
Answer
First of all, observe that
limt→0sin(xt)t=x ,
so that the integral exists as a bona fide Riemann integral. Next, given x,y∈R,
|f(x)−f(y)|≤∫π0|sin(xt)−sin(yt)|tdt.
Now use the inequality |sina−sinb|≤… to conclude that f is continuous.
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