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Re: ALL functions are continous!
Posted:
Jan 30, 2006 1:50 AM
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Let R:=(-infty,infty).
Try to prove (or disprove) following two propositions (A) and (B) :
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(A)"For every function f: R ---> R there is a dense set D ,
D being a subset of R , such that the restriction
f_{D}: D ---> R} is continuous. "
[ see: Henry Blumberg, Trans.Amer.Math.Soc., 24(1922) 113-128.]
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(B)"The set of points at which a function f:R-->R has ordinary
discontinuities is enumerable (or finite, or absent) ;
whereas the set of points of discontinuity of the second
kind may be unenumerable."
[see:
E. W. HOBSON , , The Theory of Functions of a Real Variable and the
Theory of Fourier's series, Volume I and Volume II ,Dover Publications,
Inc,New York, First edition (in one volume)
Third edition; Vol.I, 1927 , p.304 ,
as well as the works (perhaps of interest):
William Henry YOUNG , Quarterly Journal of Mathematics, vol.XXXIX,p.67
Moritz PASCH , Einleitung in die Differential und
Integralrechnung,p.139
]
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