# How do asymptotes relate to boundedness?

Oct 5, 2015

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 $\left(- \infty , a\right)$ or $\left(a , \infty\right)$, 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 $\mathbb{R}$. For example $f \left(x\right) = {x}^{3}$ 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 $f : \mathbb{R} \to \mathbb{R}$ defined as follows:

$f \left(x\right) = \left\{\begin{matrix}0 & \text{if " x " is irrational" \\ q & "if " x=p/q " in lowest terms and " q " is even" \\ -q & "if " x=p/q " in lowest terms and " q " is odd}\end{matrix}\right.$

where $p , q \in \mathbb{Z}$, with $q > 0$.

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