Example about about function has no upper and lower bound I want to find a function $f:[-1,1]$ to $\mathbb{R}$ which has no upper bound and lower bound.
Does the linear function $f(x) = \tan(x)$ work, and if so how?
Appreaciate any help with that.

Consider for example $x\mapsto \frac{x}{(x-1)(x+1)}$ (you may set an arbitrary value at $1$ and $-1$ so that it's well-defined).
If you want to stick with $\tan$, consider $g:x\mapsto \frac{\pi}2 x$. Then $\tan \circ g$ with some additional values at $x=-1$ and $x=1$ fits the bill.
