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{\bf Question}

If $f:R\rightarrow R, \,\,\, g:R\rightarrow R$ etc. discuss the
relationship between the two statements

\begin{itemize}
\item[i)]
$f$ is continuous a.e.

\item[ii)]
there is a continuous $g$ such that $f=g$ a.e.

\end{itemize}


\vspace{0.25in}

{\bf Answer}

\begin{itemize}
\item[a)]
Let $f=X_Q$ then if $g=0$, $g=f$ a.e.  So $g$ is continuous, but
$f$ is continuous nowhere.

\item[b)]
Let $f(x)=1, \,\,\, x\geq0, \,\,\,f$ is continuous a.e.

If $g=f$ a.e.  Then there is a sequence $x_n\to 0+$ such that
$f(x_n)=1$, and also a sequence $y_n\to0-$ such that $g(y_n)=0$.
Therefore $g$ is not continuous at 0.



\end{itemize}

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