For $x \ge 1,$ let $f$ be the function defined as follows: \[f(x) = \left\{ \begin{array}{cl} \lfloor x \rfloor \left| x - \lfloor x \rfloor - \dfrac{1}{2 \lfloor x \rfloor} \right| & \text{if $x < \lfloor x \rfloor + \dfrac{1}{\lfloor x \rfloor}$}, \\ f \left( x - \dfrac{1}{\lfloor x \rfloor} \right) & \text{otherwise}. \end{array} \right.\]Let $g(x) = 2^{x - 2007}.$ Compute the number of points in which the graphs of $f$ and $g$ intersect.

I've started by noting the second function intersects with the third at every integer from 1 to 2006...

 Jul 12, 2021

*sorry, i meant the first function. The second seems to intersect a few more times, but I'm not sure whether to count the intersections of each of the pieces with the third individually, or as a whole. i.e. individually, there would be around 2007*2 intersections, yet as a whole it would be around 2007.

 Jul 12, 2021

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