Abysmal floor function

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...