Is this integral less than infinity?

Question

Assume the following integral:

$$ \int\limits_{-\infty}^{\infty}\frac{f\left(x\right)} {BB\left(\lceil abs\left(x\right)\rceil\right)}\mathrm{d}x $$

Where $f\left(x\right)$ is any computable function and $BB\left(n\right)$ is the busy beaver function.

Is there a function $f\left(x\right)$ which is everywhere less than $\infty$, but this integral equals $\infty$? I think such a function does not exist, but maybe i am wrong.

Thank you

Answer 1

Hint: for $|x|$ large enough, we will have $BB(\lceil |x|\rceil)>f(x)\cdot x^2$.

Answer 2

This integral is indeed always finite, at least for every notion of "computable function on $\mathbb{R}$" that I'm aware of.

Let $b(x)=BB(\lceil abs(x)\rceil)$ for short. It's enough to show that $b$ grows fast enough - that is, for every computable $g$ there is some $n$ such that $$\forall x\in\mathbb{R}, \quad\vert x\vert>n\implies b(x)>g(x).$$

The details of this proof will vary depending exactly what formalism you use.