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

ceiling-and-floor-functions

Let $f(x) = \lfloor x \lfloor x \rfloor \rfloor$ for $x \ge 0.$

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

I tried splitting it into cases: $\lfloor x\rfloor=1$, $\lfloor x\rfloor=2$, … ,$\lfloor x\rfloor=10$, but I'm not sure how to calculate the number of $f(x)$'s. Could somebody help me here?

Thanks!!

Best Answer

Very good hints are in the comments. Please try more before looking at the solution:

$47$

Working: If you consider $g(x)=x\lfloor x\rfloor$, it is

piecewise linear as in $g(x)=nx$ if $x \in [n, n+1)$.

So $g(x)$ takes integer values $k$ times where $k = f(x)$.

To further clarify, within $[0, 1), [1, 2), [2, 3), \dots, [9, 10), \{10\}$, $h(x)$ takes the following number of values:

$1, 1, 2, \ldots, 9, 1.$

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