Unbounded function on compact interval?

Question

So what are some unbounded function on compact interval, if there is any?

Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?

Answer 1

For instance, $f:[-1,1] \rightarrow \mathbb R$ defined as $f(x)=1/x$ if $x\neq 0$ and $f(0)=0$ is defined on a compact domain $[-1,1]$ but it is not bounded.

Recall the Weierstrass theorem:

"Every continuous function on a compact domain has at least one maximum and one minimum"

So negating the above statement we obtain that:

"No maximum or minimum and the function has compact support then it must be discontinuous"

In other words, if you look for an unbounded function (in particular, no maximum or minimum) defined on a compact domain then you must look for a discontinuous function.

Answer 2

consider for example $$f:[0,1]\to \Bbb R\\ f(x) = \begin{cases} \frac 1x &\text{if} & \frac 1x\in\Bbb N, x\neq 0 \\ 0 &\text{otherwise} \end{cases}$$

which is unbounded on $[0,1]$.