Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Topic: ALL functions are continous!
Replies: 8   Last Post: Jan 30, 2006 6:53 AM

 Messages: [ Previous | Next ]
 Alex.Lupas Posts: 893 Registered: 12/6/04
Re: ALL functions are continous!
Posted: Jan 30, 2006 1:50 AM

Let R:=(-infty,infty).
Try to prove (or disprove) following two propositions (A) and (B) :

======================================================
(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.]
=======================================================

(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
]

Date Subject Author
1/29/06 NILS BÖRJESSON
1/29/06 Mike Varney
1/29/06 Virgil
1/29/06 Michael Barr
1/29/06 Mike Varney
1/30/06 Robert Low
1/29/06 Robert Israel
1/30/06 Toni Lassila
1/30/06 Alex.Lupas