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Topic: Continuity of f(x) = 1/x
Replies: 11   Last Post: May 6, 2007 6:05 PM

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John Harper

Posts: 128
Registered: 12/3/04
Re: Continuity of f(x) = 1/x
Posted: Apr 30, 2007 7:45 PM
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On Mon, 30 Apr 2007, Humberto Bortolossi wrote:

> Date: Mon, 30 Apr 2007 07:58:36 -0300
> From: Humberto Bortolossi <>
> Subject: Continuity of f(x) = 1/x
> Greetings!
> Nowadays, most calculus books say that f(x) = 1/x is discontinuous in x = 0.
> However, "analytic" oriented books (like Apostol) say f(x) = 1/x is
> continuous: the point 0 doesn't matter, since it doesn't belong to the
> function's domain.
> I'm really curious to know when and who made this bifurcation. What concept
> came first? Any references?

Some years ago I found that the history of definitions of continuity
is remarkably contorted. The following 10 textbooks, all widely used,
contain at least five essentially different definitions of continuity
of a real function of one real variable. (There may be six: Whittaker
and Watson has an ambiguity!) That the trouble arises with domains
that are subsets of the real line was already appreciated by Hedrick
in 1904 when pointing out an error of Goursat's that was repeated by
Hobson (1907).

Bartle, R.G. and Sherbert, D.R. 1992 Introduction to real analysis,
2nd ed. Wiley, New York.

Goursat, E. 1904 A course in mathematical analysis, English trans.
by E.R. Hedrick. Ginn, Boston.

Harkness, J. and Morley, F. 1893, A treatise on the theory of functions.
Stechert, New York.

Hardy, G.H. 1908 A Course of Pure Mathematics, 1st ed.
Cambridge University Press.

Hardy, G.H. 1952 A Course of Pure Mathematics, 10th ed.
Cambridge University Press.

Hobson, E.W. 1907 The Theory of Functions of a Real Variable and the
Theory of Fourier's Series. Cambridge University Press.

Jordan, M.C. 1893 Cours d'Analyse de l'Ecole Polytechnique, Tome 1,
2nd ed. Gauthier-Villars, Paris.

Pierpont, J. 1905 Lectures on the Theory of Functions of Real
Variables, Vol. 1. Ginn, Boston.

Whittaker, E.T. 1902 A Course of Modern Analysis. Cambridge
University Press.

Whittaker, E.T. and Watson, G.N. 1927 A Course of Modern Analysis,
4th ed. Cambridge University Press.

I have a PDF file to substantiate my claims above, which I shall
email to anyone who asks. It's 85K long, so I shan't post it here.

-- John Harper, School of Mathematics, Statistics and Computer Science,
Victoria University, PO Box 600, Wellington 6140, New Zealand
e-mail phone (+64)(4)463 5341 fax (+64)(4)463 5045

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