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{\bf Question}

Show that if $E$ is a subset of {\bf R} with finite measure, and
if the function $f:{\bf R}\rightarrow{\bf R}$ is defined by
$f(x)=m^*(E\cap(-\infty,x))$ then $f(x)$ is continuous.  Can you
find similar theorems in ${\bf R}^2$, or ${\bf R}^n$?


\vspace{0.25in}

{\bf Answer}

$x>a \Rightarrow 0\geq
f(x)-f(a)=m^*(E\cap(-\infty,x))-m^*(E\cap(-\infty,a))$

$\leq m^*(E\cap[x,a))\leq m^*([x,a))=x-a$

Similarly $x<a \Rightarrow 0\leq f(x)-f(a)\leq a-x$

Hence continuity.

${\bf R^2}$ by lines.

${\bf R^3}$ by planes.




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