Question

How do asymptotes relate to boundedness?

Answer 1

If a function has a vertical asymptote, then it will be unbounded above or below or both in any interval that contains the asymptote.

Explanation:

If a function has an oblique asymptote, then it will be unbounded above or below in at least one of `(-oo, a)` or `(a, oo)`, for any value of `a`.

A continuous function that is unbounded above or below or both in a finite interval has a vertical asymptote.

A continuous function need not has asymptotes in order to be unbounded in `RR`. For example `f(x) = x^3` has no asymptotes but is unbounded.

A discontinuous function does not need to have asymptotes in order to be unbounded in a finite interval. Consider the function `f:RR -> RR` defined as follows:

`f(x) = { (0, "if " x " is irrational"), (q, "if " x=p/q " in lowest terms and " q " is even"), (-q, "if " x=p/q " in lowest terms and " q " is odd") :}`

where `p, q in ZZ`, with `q > 0`.

This function is unbounded both above and below in any non-trivial interval.