How do asymptotes relate to boundedness?

1 Answer
Oct 5, 2015

Answer:

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.