Real analysis: Integrable function

[Niles]

Homework Statement

HI all.

In order to perform a Fourier transform on a function f(x), f(x) must be integrable, i.e.

$$\int_{-\infty}^{\infty}|f(x)|dx < \infty.$$

Can you confirm that this also implies that f(x) -> 0 for x -> (+/-) infinity?

 

[Cyosis]

Can you find a function that has a finite area beneath it while not being zero in infinity?

 

[Niles]

Can you find a function that has a finite area beneath it while not being zero in infinity?

No. But I am wondering if it is a sufficient conditions for the integral to be finite? E.g. f(x) = x^-1?

 

[Cyosis]

To avoid any confusion. Do you mean if f(x)->0 when x->+-infinity then the integral of f(x) over +- infinity converges? This is not true and you already gave a counter example. The function x^-1 goes to zero, but the integral of x^-1 from -infinity to infinity diverges.

 

[Niles]

Great, thanks!