A plot attempt of a fun example of a continuous nowhere differentiable function.

http://math.ucsd.edu/~lni/math140/W-example.pdf

This is taken from the Dover book: https://www.amazon.com/Functional-Analysis-Dover-Books-Mathematics/dp/0486662896

ClearAll[fn, f, points, p]

fn[x_, i_] := Round[10^i x]/10^i;
f[x_, n_] := Sum[fn[x, i],{i,0, n}] ;
points[sc_, n_] := (RandomVariate[UniformDistribution[{0,1}],n]/sc)// Sort
pairs[n_, sc_, numpoints_] := {#, f[#, n]}&/@ points[sc, numpoints]

Manipulate[
N[pairs[n,scale, pts],n] // ListPlot
,{{n,10},10, 100, 1}
, {{scale,1},1,100,1}
, {{pts,10},10,1000, 1}
]