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 ?
W.F.