***preserved for hysterical raisins****

I'm still not quite sure what you mean...

Let's restrict to x>0. You are correct: the function has extrema at x = N/k for all positive integers k. The extrema alternate between minima and maxima. The wavelength is generally defined to be the distance peak to peak (or trough to trough). So if you are standing on a peak (resp. trough) at x=N/k and you look along the negative x-axis toward the next peak (resp. trough) you will measure a wavelength of $\displaystyle \frac Nk - \frac N{k+2} = {2N\over k(k+2)}$. Now you can formally substitute N/x for k and get an expression for all x ($\displaystyle x^2N \over N(N+2x)$), but its not clear to me how valid that is. It is definitely not 1 when $\displaystyle x=\sqrt{N}$... how do you know that the wavelength should be 1 at $\displaystyle \sqrt{N}$?