The default graph on this page is a graph on the interval $[0,1]$ of the function $f(x)=\displaystyle\sum_{r_n<x} \frac{1}{2^n}$ where $r_n$ only ranges over the following finite list of rational numbers:

      1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7,
      4/7, 5/7, 6/7, 1/8, 3/8, 5/8, 7/8, 1/9, 2/9, 4/9, 5/9, 7/9, 8/9, 1/10,
      3/10, 7/10, 9/10, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 
      19/11, 10/11, 1/12, 5/12, 7/12, 11/12

This $f(x)$ approximates the function that uses the full list of rational numbers between 0 and 1 to within about $10^{-15}$. The exact function has a jump discontinuity at every rational number but is continuous at every irrational number.

You can elect to randomly reorder the list of rational numbers and see the graph that results, and you can add a lot of extra fractions before randomizing, which allows jumps at more places. Just click one:

Use default list      Randomly order default list

Add 1/199, 2/199, ..., 198/199 and randomize

You can modify the function to $f(x)=\displaystyle(z-1)\sum_{r_n<x}\frac{1}{z^n}$ for $z$ in the range 1.1 to 2. When using the full list of rational numbers, this function will still satisfy $f(1)=1$. For $z<2$, the jumps will be smaller than they are for $z=2$, so you can see more of them.

z = 2

Canvas not supported.