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I want to find a example of Borel measurable function which is not continuous. I think that it is a simple or step function or semicontinuous function. Please help me for find it.

Thanks.

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You already mention some in your post. What else do you want? –  user27126 Apr 27 '13 at 7:46
    
An "even less" continuous example would be the characteristic function of the rationals. –  Hagen von Eitzen Apr 27 '13 at 7:58
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up vote 1 down vote accepted

The indicator function on $\mathbb Q$ is borel measurable but is nowhere continuous.

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