Finding the number of possible values of $\lfloor x \lfloor x \rfloor \rfloor$ for $0 \le x \le 10.$

Question

Find the number of possible values of $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $0 \le x \le 10$.

I tried to see if I could set up some inequality, or do some casework, but I still can't get any progress.

Answer 1

Clearly the maximum is $100$ when $x=10$ and when $x=9$ you have $f(x)=81$. In between there, the inner floor always gives $9$. Can you convince yourself that there is an $x$ that gives all the values $82-99$? Then apply that to the other intervals.