[Math] Bounded function in $[0,1]$ without max and min.

Question

Do we have a function $f$ defined in $[0, 1]$, which is bounded but has no maximum and minimum ?

I do know that $\arctan x$ can give me a hint, which is bounded without max and min but that's in $\mathbb R$.
Thanks!

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EDITED:
Firstly thank all the answers that I received.
I would like to ask a little additional question: could we find some edited or mixed trigonometric functions to satisfy this problem?
If yes, a example please!
The reason is, that I got stuck with the idea of $\arctan x$ and other possible trigonometric functions.

Thanks!

Answer 1

$f(x)=x$ if $.25<x<.75$, and $f(x)=0.5$ otherwise