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(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?

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    but how do you proof the range of f is a second category subset of R2?2011-08-04
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    @mathabc: Mike’s function is a surjection: its range is *all* of $\mathbb{R}^2$.2011-08-04
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    Whoops, you're right that doesn't work.2011-08-04
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    my mistake,what I mean is how to proof the graph of f is a second category subset of R2?2011-08-04

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