(1)Please give a real-valued function $f$ satisfies the set $\{(x,f(x)):x\text{ belongs to }\mathbb{R}\}$ is a second category subset of $\mathbb{R}^2$? (2)Please give a real-valued function f satisfies the set $\{(x,f(x)):x\text{ belongs to }\mathbb{R}\}$ is a non-measurable set in $\mathbb{R}^2$ in the Lebesgue sense?
Existence of Real-Valued Functions Satisfying Certain Properties
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real-analysis
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0but how do you proof the range of f is a second category subset of R2? – 2011-08-04
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0@mathabc: Mike’s function is a surjection: its range is *all* of $\mathbb{R}^2$. – 2011-08-04
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0Whoops, you're right that doesn't work. – 2011-08-04
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0my mistake,what I mean is how to proof the graph of f is a second category subset of R2? – 2011-08-04