Improving that tikzpicture pgfplot by breaking the domain

Question

I'm trying to improve that figure:

\documentclass{standalone}
\begin{document}
\usepackage{tikz}
\usepackage{pgfplots}
\begin{tikzpicture}
  \begin{axis} [axis lines=center, height=46mm, width=120mm, ymin = 0.8499]
    \addplot [domain=0:1, samples=3333, thick] { 1-(1-x)^floor(2/x) };
  \end{axis}
\end{tikzpicture}
\end{document}

I'd like to

• make the curve discontinuous: the current down-going lines are pure artifact
• avoid the non-existing transititon there is near x=0.1
• if possible, make it faster. I'm ready to make a black triangle for x<1/30 or so.

I think an explicit loop drawing multiple curves in range ]1/(n+1),1/n[ is the way to go; but how?

Answer 1

Perhaps something like this? It's not very fast, but you can reduce the samples or the \foreach limit of 500.

\documentclass{standalone}
\usepackage   {pgfplots}
\pgfplotsset  {compat=1.17}

\begin{document}
\begin{tikzpicture}[line cap=round]
\begin{axis} [axis lines=center, height=45mm, width=120mm, xmin=0, ymin=0.8499]
  \foreach\i in {2,...,500}
  {%
    \addplot[domain={2.001/(\i+1)}:{2/\i}, samples=11,smooth] {1-(1-\x)^floor(2/\x)};
  }
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

This is just to show that it is possible to introduce jumps in a single plot:

\documentclass[tikz, border=1 cm]{standalone}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
  \begin{axis} [axis lines=center, height=46mm, width=120mm, ymin = 0.8499]
    \addplot [
    domain=0:1,
    samples=3333,
    thick, 
    unbounded coords=jump, line join=bevel,
    x filter/.expression={
        x>0.08 && floor(2/x)+0.01/x>2/x ? nan : x}
                ] { 1-(1-x)^floor(2/x) };   
  \end{axis}
\end{tikzpicture}
\end{document}