Re: [HM] An odd (mathematical) remark by GHH
Gordon Fisher (gfisher@shentel.net)
Wed, 18 Nov 1998 17:45:31
At 02:03 PM 11/18/98 +0100, Walter Felscher wrote:
>
> Professor J.F.Harper in his contribution "History of def. of continuity",
> from November 17th, has drawn our attention to an odd remark by the great
> G.H.Hardy.
>
> Mr. Harper quotes from Hardy's "A Course of Pure Mathematics" 10th ed
> Cambridge 1952 , p186 , first the definition of continuity and then, from
> p.189
>
> The function which is equal to 1 when x is rational and to 0 when x is
> irrational (Ch. II, Ex. XVI. 10) is discontinuous for all values of x.
> So too is any function which is defined only for rational or for
> irrational values of x."
>
> [In order to verify this quotation, I have taken the pains to waddle to
> the mathematics library here which does not have the 10th but the 4th
> edition of Hardy's book. Mr. Harper's quotation there occurs verbally as
> the second in the list of "Examples" on p.176. ]
>
> This "So too .." sentence of Hardy's is very odd indeed, because (i)
> while it is not quite clear what the "so" refers to and (ii) it suggests
> to be read to imply that the restriction of Dirichlet's function to
> rational arguments only, or to irrational arguments only, be discontinuous !
>
> We can be safe to assume that GHH did not want draw to such asinine
> conclusion, but what then did he mean by the "So too .. " sentence ? Has
> anyone ever commented on this ?
Maybe it's a question of what topology one uses? E.g., the restriction of
the Dirichlet function to rationals on a real interval may be considered to
be discontinuous in the usual topology on the real line because it fails to
be defined on intervals of the usual topology, even though it will be
continuous on the topology induced on the rationals by intersecting
intervals of the usual topology with the rationals. ???
Gordon Fisher gfisher@shentel.net