Question

What are some examples of unbounded functions?

Answer 1

Here are four examples...

Explanation:

x
The simplest example of an unbounded function is `f(x) = x`, which is unbounded for `x in (-oo, oo)`

1/x
The function `f(x) = 1/x` is unbounded on any interval that includes `x = 0`, due to a simple pole at `x = 0`.

tan(x)
The function `f(x) = tan(x)` is unbounded on any interval that includes an `x` of the form `pi/2 + npi`, since it has a vertical asymptote at each of these values.

Unbounded on any interval

Consider the function:

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

where `p` and `q` are integers and `q > 0`

This is unbounded on any open interval, since in any open interval you can find a rational number with an arbitrarily large odd or even denominator when expressed in lowest terms.