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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? –  mathabc Aug 4 '11 at 1:11
    
@mathabc: Mike’s function is a surjection: its range is all of $\mathbb{R}^2$. –  Brian M. Scott Aug 4 '11 at 1:17
    
Whoops, you're right that doesn't work. –  Mike Aug 4 '11 at 1:18
    
my mistake,what I mean is how to proof the graph of f is a second category subset of R2? –  mathabc Aug 4 '11 at 1:19
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See Gelbaum and Olmsted, Counterexamples in Analysis, Chapter 10, Plane Sets, Example 23, A real-valued function of one real variable whose graph is a nonmeasurable plane set. It's a little too long for me to type out, and it depends on Example 21, A nonmeasurable plane set having at most two points in common with any line. Example 21 takes two full pages in the book.

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Thank you very much!For(1),do you have any ideas? –  mathabc Aug 4 '11 at 1:21
    
You may use space-filling curves, aka Peano curves. –  gary Aug 4 '11 at 1:59
    
For the first, wouldn't using a combination using a nonmeasurable subset of $\mathbb R^2$ help? –  gary Aug 4 '11 at 2:02
    
@gary, OP wants the graph of a function, which a Peano curve is not. Also, I don't know what "a combination using a nonmeasurable subset" means. –  Gerry Myerson Aug 4 '11 at 3:53
    
I meant somehow using the fact that the characteristic function of a nonmeasurable set is nonmeasurable. –  gary Aug 4 '11 at 4:41
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