- #1

laonious

- 9

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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!