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Let $f$ be a real valued function then the set where $f$ is continuous is measurable.

Note the statement does not even requires that $f$ is measurable.

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    Hint: $f$ is continuous at $x$ if and only if $x\in \cap_{n=1}^\infty O_n$, where $O_n=\{x | \text{ there is a }\delta_x>0\text{ such that } |f(y)-f(z)|.2012-02-24

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