How to avoid orphan or widow when using tcolorbox?

With the version 2.02 you need the following modification. I can think about a new implementation step to reduce the overlapping.

\makeatletter
\def\tcb@split@lower@box{%
  \vfuzz=\maxdimen\vbadness=10000%
  \setbox\tcb@lowerbox=\vsplit\tcb@totallowerbox to\tcb@split@dim%
  \setbox\tcb@lowerbox=\vbox{\unvbox\tcb@lowerbox}%
  \vbadness=\tcb@thevbadness\vfuzz=\tcb@thevfuzz%
  \tcbdimto\tcb@h@lower{\ht\tcb@lowerbox+\dp\tcb@lowerbox}%
  \ifdim\tcb@h@lower>\tcb@split@dim\relax%
    \setbox\tcb@lowerbox=\vbox to\tcb@split@dim{\unvbox\tcb@lowerbox}%
  \fi%
}
\def\tcb@split@L{%
  \stepcounter{tcbbreakpart}%
  \tcb@following@title%
  \tcb@comp@h@page%
  \let\tcb@h@upper=\tcb@zpt%
  \tcbdimto\tcb@h@lower{\ht\tcb@totallowerbox+\dp\tcb@totallowerbox}%
  \tcbdimto\tcb@h@total{\tcb@h@after+\tcb@h@padding@last+\tcb@h@padtitle+\tcb@h@lower}%
  %
  \ifdim\tcb@h@total<\tcb@h@page\relax% no break
    \tcb@split@setstate@last%
    \setbox\tcb@lowerbox=\vbox{\begingroup\color{\kvtcb@collower}\unvbox\tcb@totallowerbox\endgroup}%
    \tcb@sdraw@L%
    \let\tcb@split@next=\relax%
  \else
    \tcb@split@setstate@middle%
    \tcbdimto\tcb@split@dim{\tcb@h@page-\tcb@h@padding@middle-\tcb@h@padtitle}%
    \tcb@split@lower@box%
    \tcb@sdraw@L%
    \setbox\tcb@totallowerbox=\vbox{\unvbox\tcb@totallowerbox}
    \ifdim\wd\tcb@totallowerbox=0pt
      \let\tcb@split@next=\relax%
    \else%
      \let\tcb@split@next=\tcb@split@L%
    \fi%
  \fi%
  \tcb@split@next%
}
\makeatother

---

OLD ANSWER

The splitting algorithm of tcolorbox has some lacks. E.g. there is no test if any box is empty, the vbox doesn't end without removing unnecessary space. I didn't analyse the complete algorithm.

A quick fix would be:

\documentclass{article}
\usepackage[a6paper,vmargin=15mm,hmargin=5mm]{geometry}
\usepackage{times}
\usepackage{helvet}
\usepackage{courier}
\usepackage[T1]{fontenc}
\usepackage{mathtools}
\allowdisplaybreaks
\usepackage{tcolorbox}
\usepackage{microtype}


\tcbuselibrary{breakable}
\newcounter{exercise}[subsection]
\@addtoreset{exercise}{section}


\renewcommand{\theexercise}{%
    \ifnum\value{subsection}>0\relax
        \thesubsection
    \else
        \thesection
    \fi
    .\arabic{exercise}%
}

\newenvironment{exercise}
{\par\smallskip\refstepcounter{exercise}
\begin{tcolorbox}[breakable,title=Exercise \theexercise,left=0mm,lefttitle=2mm]\ignorespaces}
{\end{tcolorbox}\par\smallskip\ignorespacesafterend}
\makeatletter
\def\endtcb@vbox{%
\unskip
\par\unskip%
\vskip\lastskip%
\par\strut\par\unskip\unskip
\setbox0=\lastbox
\vskip-.3\baselineskip
\endgroup}

\def\tcb@split@first{%
  \setcounter{tcbbreakpart}{1}%
  \ifx\kvtcb@float\@empty%
    \def\tcb@before@first{\nointerlineskip\kvtcb@beforebox}%
    \def\tcb@before@middle{\nointerlineskip\noindent}%
    \let\tcb@before@last=\tcb@before@middle%
    \long\def\tcb@after@first{\parskip\z@\par\pagebreak\nointerlineskip{\topskip\z@\null}}%
    \let\tcb@after@middle=\tcb@after@first%
    \let\tcb@after@last=\kvtcb@afterbox%
  \else%
    \edef\float@start{\noexpand\@float{tcbfloat}[\kvtcb@float]}%
    \let\tcb@before@first=\float@start%
    \let\tcb@before@middle=\float@start%
    \let\tcb@before@last=\float@start%
    \let\tcb@after@first=\end@float%
    \let\tcb@after@middle=\end@float%
    \let\tcb@after@last=\end@float%
  \fi%
  %
  \edef\kvtcb@skin@@first{\kvtcb@skin@first}%
  \edef\kvtcb@skin@@middle{\kvtcb@skin@middle}%
  \edef\kvtcb@skin@@last{\kvtcb@skin@last}%
  %
  \let\tcb@h@breaksep@first=\kvtcb@bottomsep%
  \tcbdimto\tcb@h@breaksep@middle{\kvtcb@bottomsep+\kvtcb@topsep}%
  \let\tcb@h@breaksep@last=\kvtcb@topsep%
  \tcbdimto\tcb@h@padding@first{\kvtcb@bbtop@stand+\kvtcb@bbbottom@break+\kvtcb@top@rule@stand+\kvtcb@bottom@rule@break+\tcb@h@breaksep@first+\kvtcb@boxsep*2+\kvtcb@top+\kvtcb@bottom}%
  \tcbdimto\tcb@h@padding@middle{\kvtcb@bbtop@break+\kvtcb@bbbottom@break+\kvtcb@top@rule@break+\kvtcb@bottom@rule@break+\tcb@h@breaksep@middle+\kvtcb@boxsep*2+\kvtcb@top+\kvtcb@bottom}%
  \tcbdimto\tcb@h@padding@last{\kvtcb@bbtop@break+\kvtcb@bbbottom@stand+\kvtcb@top@rule@break+\kvtcb@bottom@rule@stand+\tcb@h@breaksep@last+\kvtcb@boxsep*2+\kvtcb@top+\kvtcb@bottom}%
  %
  \tcb@split@setstate@first%
  %
  \splittopskip=0pt%
  \splitmaxdepth=0pt%
  \edef\tcb@thevfuzz{\the\vfuzz}%
  \edef\tcb@thevbadness{\the\vbadness}%
  %
  \setbox\tcb@totalupperbox=\vbox{\unvbox\tcb@upperbox}%
  \setbox\tcb@totallowerbox=\vbox{\unvbox\tcb@lowerbox}%
  %
  \tcbdimto\tcb@split@dim{\tcb@h@page-\tcb@h@before-\tcb@h@padding@first-\tcb@h@padtitle}%
  \ifdim\tcb@split@dim<\tcb@h@upper\relax%
    \tcb@split@upper@box%
    \tcb@sdraw@U%
    \let\tcb@split@next=\tcb@split@USL%
  \else%
    \setbox\tcb@upperbox=\vbox{\unvbox\tcb@totalupperbox}%
    \tcbdimto\tcb@split@dim{\tcb@split@dim-\tcb@h@upper-\tcb@h@segment}
    \ifdim\tcb@split@dim<\kvtcb@breakminlines\baselineskip\relax%
      \tcb@sdraw@U%
      \let\tcb@split@next=\tcb@split@SL%
    \else
      \tcb@split@lower@box%
      \tcb@sdraw@USL%
       \ifvoid\tcb@totallowerbox
          \let\tcb@split@next=\relax%
       \else
          \let\tcb@split@next=\tcb@split@L%
       \fi%

    \fi%
  \fi%
  \tcb@split@next%
}



\makeatother

\begin{document}
\section{Function}
The notation $f(x)$ can be interpreted as 
\begin{itemize}
    \item an algebraic expression in a variable\footnote{variable is \ldots} $x$ or
    \item the value of function $f$ when its variable  equals to $x$.
\end{itemize}
\begin{exercise}
Explain what  each of the following means
\begin{enumerate}
    \item $h(x)$
    \item $h(x+h)$
    \item $g(a)$
    \item $g(a+b-c)$
\end{enumerate}
\tcblower
\begin{enumerate}
    \item $h(x)$ \ldots
    \item $h(x+h)$ \ldots
    \item $g(a)$ \ldots
    \item $g(a+b-c)$ \ldots
\end{enumerate}
\end{exercise}
Note that $f(x)$ does not mean $f$ multiplied by $x$.
\end{document}

I am the author of tcolorbox and I have just uploaded version 2.22 to CTAN. This new version removed (some/most) of the addressed problems.

1. "There are unwanted blank pages"

   I found the cause of the problem and (hopefully) eliminated it. Since the breaking of the upper part / lower part features of tcolorbox is sometimes a little bit tricky, I may have overlooked something. Let me know, if blank pages occur despite of the corrections.

2. "The inconsistent color scope occurs"

   I found and (hopefully) corrected a color problem which caused one inconsistent color scope effect (the \item thing). Nevertheless, a color change inside the text content does not survive the breaking. This is a known drawback for tcolorbox. Im am sorry to say that I have no real solution in mind for this. But as far as I know it is a common problem when splitting \vboxes.

3. "There is an unwanted orphan/widow effect"

   Marco Daniel provided a quick solution (thank you for that!) which hides one of the void boxes. Version 2.22 tries to circumvent all situations where an empty box appears at the end of a break sequence. I've done a lot of tests with the new version and all worked fine. So, hopefully, this effect should not happen again - let me know, if it does ...