I'm a university student studing math and I have been given the question above.
I've got an idea of whats going on. It is clear that the $f(x)$ will cycle through in the range $[0,1)$. I'm not sure why the question is putting so much emphasis on $[0,2]$ because I only see the function defined in the aforementioned range $[0,1)$. As for a maximum wouldn't it be $1$ ?
If anyone has any tips or insights that I might be missing I would love to hear it!
Thank you for your time!
You can define this function on every interval. I guess the question is to show a counter-example of the statement "$f$ bounded on a compact interval $[a,b]$ implies $f$ has a maximum".
Show first that for all $x \in [0,2]$, you have $f(x) < 1$. It is not so hard. Thus, show that there exists a sequence $x_n \in [0,2]$ for which $f(x_n) \to 1$ : it will show that $f$ has for supremum $1$, but that it is not attained. For example, $x_n = 1 - \frac{1}{n+1}$ will work.