What are some examples of unbounded functions?

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What are some examples of unbounded functions?

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Answer 1 · 2015-09-09T07:01:52

Here are four examples...

Explanation:

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

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

tan(x)
The function $f (x) = tan (x)$ $f(x) = tan(x)$ is unbounded on any interval that includes an $x$ $x$ of the form $\frac{π}{2} + n π$ $pi/2 + npi$ , 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.

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What are some examples of unbounded functions?

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