Let $f$ be a measurable real-valued function on an interval $[a,b]$. Prove that given $\delta>0$ there is a continuous function $\phi$ on $[a,b]$ such that $m(\{x: f(x) \ne g(x)\}) < \delta$. Is the same true for a function $f$ defined on $\mathbb R$?
Prove there exists a continuous function
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real-analysis
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0Martini, can you please provide a decomposition of R into the intervals you are talking about? I got the first part down by heart. But I'm still very confused by the second. Thank you! – 2012-05-20