Problem: prove that the set of $C([0, 1])$ functions whose derivative is defined at every point (and it is either finite or infinite) is of the first Baire category.
I have no idea how to approach this and would be very grateful for any help.
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2I removed your last sentence because it is asking a separate question. – Nate Eldredge May 26 '14 at 16:03
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oo, exactly, thank you, I thought so that the Lagrange formula approach – Demons94 May 26 '14 at 16:43
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@Chris Janjigian: Your Lipschitz condition comment doesn't take care of and it is either finite or infinite. (Maybe this part was edited in after you made your comment, however.) Also, being finitely differentiable at a point doesn't imply being Lipschitz on an interval. Consider $x^2$ times the characteristic function of the rationals at $x=0.$ (Continuous examples also exist by constructions involving things like $f(p/q) = 1/q^{2},$ defined to be zero at irrationals.) – Dave L. Renfro May 27 '14 at 16:25
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@DaveL.Renfro Yes thanks, I definitely did not think that through carefully. – Chris Janjigian May 27 '14 at 17:32
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@Demons94 sorry for the bad hint. – Chris Janjigian May 27 '14 at 17:39