An Interesting Function

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Let left{ r_1,r_2,r_3, dots right}  be any enumeration of the rational numbers. Define the function fleft( x right) =sum_{r_n le x} frac{1} {2^n}. Then f has the following properties:

  1. 0<fleft( x right) <1
  2. fleft( x right) is strictly increasing, and thus has a derivative almost everywhere.
  3. fleft( x right) <frac{1}{2^k} Leftrightarrow x<minleft{ r_1,r_2,dots,r_k right}  for k=1,2,3,dots
  4. r_k < x Rightarrow frac{1}{2^k} < fleft( x right) for k=1,2,3,dots
  5. If we define f_Nleft( x right) =sum_{r_n le x,nle N} frac{1} {2^n}, then we have for N>0 
f_Nleft( x right) le f_{N+1}left( x right) and
f_Nleft( x right) < fleft( x right)<f_Nleft( x right) +frac{1}{2^N}