It's unlikely that Hardy's Pure Math. was the first text to use
the modern definition of point wise continuity, though the
convention is to interchange use of the symbols for epsilon
and delta.
The text by Bartle and Sherbert is a standard undergraduate
intro. to adv. calc. It should have the same definition earlier
than page 140 (I don't have a copy).
Their definition using neighborhoods is a geometric definition,
an analog to the analytic one given by Hardy. It states that
the neighborhood V [c; delta] is mapped under f into the
neighborhood V [f(c); epsilon]. These definitions are equivalent.
The analytic definition generalizes to metric spaces. Simply
rewrite "f is con't. at pt. a if for all e > 0, there exists d > 0
s.t. || x - a || < d whenever || fx - fa || < e ". Note that the
norm || , || defines the metric of interest.
Certainly the growth of work in functional analysis during the
20's and 30's was more appropriately represented using the
geometric concepts. After all, now we're potentially dealing
with spaces of arbitrary dimension. Hardy's book summed
up basic analysis of the end of the 19th Century.
So while I don't know "who" changed (or generalized) the
definition, maybe the comment can help narrow down the
search ! My guess is Frechet or Hausdorff.
Al Barron
> ----------
> From: John F Harper[SMTP:John.Harper@vuw.ac.nz]
> Sent: Monday, November 16, 1998 8:07 PM
> To: historia-matematica@chasque.apc.org
> Subject: [HM] History of def of continuity
>
> I would like to know who changed the definition of continuity, when,
> and why. The details herewith will show that the definition appears to
> have been changed during the present century.
>
> 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