# Existence of Real-Valued Functions Satisfying Certain Properties

(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 F 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

For the first, wouldn't using a combination using a nonmeasurable subset of $\mathbb R^2$ help? –  gary Aug 4 '11 at 2:02