Hardy's "A Course of Pure Mathematics" 10th ed Cambridge 1952 p186 says:
phi(x) is continuous for x = xi if, given delta, we can choose
epsilon(delta) so that |phi(x)-phi(xi)| < delta if
0 <= |x-xi| <= epsilon(delta).
He goes on to make an exception for continuity at an end-point of an
interval (a,b) on which phi is defined. (For Hardy, (a,b) means the closed
interval; see p30.) He says (still on p186):
In this case it is convenient to make a slight and natural change ...
phi(x) is continuous for x = a if phi(a+0) exists and is equal to
phi(a) ...
He is thus heading in the direction of the modern definition but stops
before reaching it, as will become obvious in my next quotation (p189):
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."
Now consider B+S = Bartle & Sherbert's "Introduction to Real Analysis" 2nd
ed Wiley 1992 p.140 (I picked this book because it gets there without
going through general metric spaces on the way.) Let R be the set of real
numbers, and let V_a(b) be the set of x in R such that |x-b| < a.
Silently expanding some abbreviations to avoid a lot of LaTeX, I quote:
Let A be a subset of R, let f: A -> R, and let c be a point of A. We
say that f is continuous at c if, given any neighbourhood
V_epsilon(f(c)) of f(c) there exists a neighbourhood V_delta(c) of c
such that if x is any point of the intersection of A and V_delta(c),
then f(x) belongs to V_epsilon(f(c)).
The function f(x) which = 0 on the rationals, and is defined only on the
rationals, is continuous nowhere according to Hardy, but is continuous for
rational c according to B+S, because A = Q, the set of rationals. If Hardy
had been using B+S notation he would presumably have omitted
"the intersection of A and" in their definition (except when A was a
closed interval).
I suspect the old definition was abandoned because it does not generalise
so well to more general metric spaces. I do not say Hardy invented the old
definition or B+S the new one; I used their books merely as convenient
statements of the different definitions.
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail john.harper@vuw.ac.nz phone (+64)(4)471 5341 fax (+64)(4)495 5045