Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I am studying past analysis prelim exams to take in the fall and have run into one which really has me stumped:

Let f be a real-valued Lebesgue integral function on [0,\infty).

Define

F(x)=\int_{0}^{\infty}f(t)\cos(xt)\,dt.

Show that F is defined on R and is continuous on R.

Show that \lim_{\rightarrow \infty}F(x)=0.

That F is defined is straightforward I think, the part I am struggling with is showing that it is continuous.

Going the route of a direct proof:

F(x)-F(y)$=\int_{0}^{\infty}f(t)(\cos(xt)-\cos(yt))\,dt,

but it isn't clear to me how to show that this is enough to make the integral small.

Any ideas would be greatly appreciated, thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Analysis Prelim prep: Lebesgue integration

**Physics Forums | Science Articles, Homework Help, Discussion**