According to the documentation, the breakable flag of tcolorbox should create two partialy boxed parts, that is, the first page part has no bottomrule and the second page part has no toprule.
Unfortunately, I do not understand why my example yields two parts with two completely boxed parts. What am I missing here?
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{natbib}
\usepackage{graphicx}
\usepackage{lipsum}
\usepackage{tcolorbox}
\usepackage{amsmath}
\tcbuselibrary{breakable}
\newcounter{myexamplectr}
\newenvironment{example}[1][]
{\refstepcounter{myexamplectr} \begin{tcolorbox}[boxrule=0.5mm, arc=0mm, breakable=true, #1]
\texttt{Example \themyexamplectr}: }{\end{tcolorbox}}
\begin{document}
\lipsum[1]
\begin{example}[before skip=1cm,after skip=1cm]
\lipsum[1-6]
\end{example}
\lipsum[1]
\end{document}