$\def\R{\mathbb R}$Let $A= \{x: f \text{ is continuous at $x$}\}$ for $f : \R\to \R$ , why is $A$ Borel measurable?
Why Continuity set is a borel set?
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measure-theory
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0Think about $\liminf$ and $\limsup$ – 2012-09-18
2 Answers
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The set $A$ is what is called a $G_\delta$ set--a countable intersection of open sets. Since open sets are Borel measurable, and since countable intersections of measurable sets are measurable, then $A$ is measurable.
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Let $A_{a,b}=\{x\mid \exists\delta>0\colon\forall y\in (x-\delta,x+\delta)\colon a
Then $A_{a,b}$ is open (why?) and $A=\bigcap_{n\in \mathbb N}\bigcup_{q\in \mathbb Q}A_{q,q+\frac1n}$ (again: why?) is a countable intersection of open sets.
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0Perfect! So do you thin$k$ it is still the case for A={x:x is differentiable} - I learnt that most of these measurable problems are concerned with findin$g$ a lim-sup and liminf but what would it be about differentiables? – 2012-09-19