I would suggest you to use \newmdtheoremenv (from the mdframed package) to define your environment; using middleextra and secondextra you can easily achieve what you want; a little example (the legend "Example (Cont.)" will appear boldfaced and framed, but of course, you can choose the formatting that best suits your needs):
\documentclass{article}
\usepackage[paperwidth=12cm,paperheight=10cm]{geometry}% JUST FOR THE EXAMPLE
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\newmdtheoremenv[
hidealllines=true,
innerleftmargin=0pt,
innerrightmargin=0pt,
middleextra={%
\node[draw,anchor=west,yshift=0.7cm] at (P-|O) {\bfseries Example~\theexample~(Cont.)};
},
secondextra={%
\node[draw,anchor=west,yshift=0.7cm] at (P-|O) {\bfseries Example~\theexample~(Cont.)};
}
]{example}{Example}
\begin{document}
\begin{example}
\lipsum[1-3]
\end{example}
\begin{example}
\lipsum[1-3]
\end{example}
\end{document}
Of course, one can use a predefined style or define a own theorem style using \newtheoremstyle; here's an example using the settings for myexamplestyle as given in the question:
\documentclass{article}
\usepackage[paperwidth=12cm,paperheight=10cm]{geometry}% just for the example
\usepackage{amsthm}
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\newtheoremstyle{myexamplestyle}
{}{}{}{}{\bfseries}{.}{ }{}
\theoremstyle{myexamplestyle}
\newmdtheoremenv[
hidealllines=true,
innerleftmargin=0pt,
innerrightmargin=0pt,
middleextra={%
\node[anchor=west,yshift=0.7cm,inner xsep=0pt] at (P-|O) {\bfseries Example~\theexample~(Cont.)};
},
secondextra={%
\node[anchor=west,yshift=0.7cm,inner xsep=0pt] at (P-|O) {\bfseries Example~\theexample~(Cont.)};
}
]{example}{Example}
\begin{document}
\begin{example}
\lipsum[1-3]
\end{example}
\begin{example}
\lipsum[1-3]
\end{example}
\end{document}
And here's another option using this time the thmtools package as a front-end for amsthm; using this approach, you define a style (compatible with the amsthm definitions) that can be applied to as many theorem-like structures (numbered or unnumbered) as you like. In the example below I show the style and how to use it for a numbered structure and for an unnumbered one; I also show how one can modify attributes for the structures, but keeping the "Continuation" style:
\documentclass{article}
\usepackage[paperwidth=12cm,paperheight=10cm]{geometry}% just for the example
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\makeatletter
\declaretheoremstyle[
headfont=\normalfont\bfseries,
notefont=\mdseries, notebraces={(}{)},
bodyfont=\normalfont,
preheadhook={
\begin{mdframed}[hidealllines=true,
innerleftmargin=0pt,
innerrightmargin=0pt,
middleextra={\node[anchor=west,yshift=2ex,inner xsep=0pt] at (P-|O) {\bfseries\thmt@thmname~\csname the\thmt@envname\endcsname
~(Cont.)};},%
secondextra={\node[anchor=west,yshift=2ex,inner xsep=0pt] at (P-|O) {\bfseries \thmt@thmname~\csname the\thmt@envname\endcsname~(Cont.)};}%
]},
prefoothook=\end{mdframed}
]{myexamplestyle}
\makeatother
\declaretheorem[style=myexamplestyle,numbered=no]{example}
\declaretheorem[style=myexamplestyle,postheadhook=\itshape]{remark}
\begin{document}
\begin{example}
\lipsum[1-3]
\end{example}
\begin{remark}
\lipsum[1-3]
\end{remark}
\end{document}
And another option using \surroundwithmdframed; using this approach the theorems and their styles are defined in the usual way through amsthm (or ntheorem) and the resulting environments are given the desired "Continuation" style with the help of mdframed:
\documentclass{article}
\usepackage[paperwidth=12cm,paperheight=10cm]{geometry}% just for the example
\usepackage{amsthm}
\usepackage[framemethod=tikz]{mdframed}
\usepackage{lipsum}
\newtheoremstyle{myexamplestyle}
{}{}{}{}{\bfseries}{.}{ }{}
\theoremstyle{myexamplestyle}
\newtheorem*{myexample}{Example}
\surroundwithmdframed[
hidealllines=true,
innerleftmargin=0pt,
innerrightmargin=0pt,
middleextra={\node[anchor=west,yshift=2ex,inner xsep=0pt] at (P-|O) {\bfseries Example~(Cont.)};},%
secondextra={\node[anchor=west,yshift=2ex,inner xsep=0pt] at (P-|O) {\bfseries Example~(Cont.)};}%
]{myexample}
\begin{document}
\begin{myexample}
\lipsum[1-3]
\end{myexample}
\end{document}
