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I'm not sure these two statement are not same thing.

$$ "f equals a continuous function a.e." & "f is continuous a.e."$$

The concept is too much abstract, so I wanna find some counter examples.

  1. a function $f$ and a continuous function $g$ s.t. $f=g$ a.e and $f$ is not continuous a.e.

  2. a function $f$ continuous a.e. s.t. there exists no continuous function $g$ with $f=g$ a.e.

In addition I wanna find a Riemann integrable function which has an uncountable set of discontinuities.

Is there some nice example to understand those concepts, note that please.

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up vote 10 down vote accepted

Function continuous a.e. but not equal a.e. to a continuous function: $f(x) = 0$ if $x < 0$ and $f(x) = 1$ if $x \geq 0$

Function equal a.e. to a continuous function but not continuous a.e.: $f$ = the characteristic function of the rationals.

Riemann integrable function with discontinuities on an uncountable set: $f$ = the characteristic function of the Cantor set in $[0,1]$. Let $C$ be the cantor set. Then $C$ is perfect so it is uncountable and closed. Hence $f$ is continuous on the complement of $C$ because these points are all interior points. But since $C$ contains no open interval, every point of $C$ is a boundary point, so $f$ is discontinuous on $C$. $C$ has measure zero, so $f$ is Riemann integrable.

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Those are simple and great example. Thanks for your help! – wowhapjs Jun 10 '12 at 12:45

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