Can anyone give me an example of a function $f\colon[a,b]\to\mathbb{R}$ that is unbounded?
Example of a function $f: [a, b] \to \mathbb{R}$ that is unbounded.
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2Only if there are answers which are any good! He's only had six questions, it's a bit soon to be drawing conclusions. – 2012-07-04
3 Answers
How about the function whose value is $\tan(x)$ where this is defined, and $0$ whenever $\tan (x)$ is undefined? Just go for the portion from $[0,\pi]$ to ${\Bbb R}$ if you like.
Consider the interval $[0,1]$. Define
$f(x) = \begin{cases} \frac{1}{x} & \quad x \neq 0 \\ 0 & \quad x = 0 \end{cases}$
Clearly $f$ is unbounded. Note that $f$ is not continuous.
There can be no continuous function that has that property (since a continuous function defined on a finite closed interval must achieve a maximum and a minimum value, hence must be bounded; or, to use higher power results, the image of a compact set under a continuous function is compact). However, you can define discontinuous functions that do rather easily. E.g, $f(x) = \left\{\begin{array}{ll} \frac{1}{x-a} - \frac{1}{b-x}&\text{if }x\neq a\text{ and }x\neq b\\ 0 &\text{if }x=a\text{ or }x=b. \end{array}\right.$ This function has $\lim\limits_{x\to a^+}f(x) = \infty$ and $\lim\limits_{x\to b^{-}}f(x)=-\infty$.