Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

400 MATHEMATICAL PHILOSOPHY
for the slope of the tangent at any point whose abscissa
dy
x yields a definite value for -. But at the point, x =0
y has no value; it is indeed meaningless as involving
dx
division by zero. Hence the curve, though it is continuous at the point in question, does not admit a tangent
at the point.
As you may know, such curves are infinitely numerous.
For other examples one may consult a memoir on discontinuous functions by Darboux in the Annales de l'École
Normale superieure, Vol. IV, 2d series. Some examples
of the kind are beautifully discussed in the appendix
of W. B. Smith's Infinitesimal Analysis. See also W. K.
Clifford's remarks on " Crinkly Curves " in Vol. I of his
Elements of Dynamic. The matter is, of course, dealt
with in all up-to-date books of advanced calculus or of
the theory of functions of the real variable. By a famous
example adduced by Weierstrass, it was shown that a
curve may be continuous at every point and yet have no
tangent at any point.
Phenomena of the kind above indicated naturally
raise another question, which I may state without attempting to discuss it here. The question is: Is it possible to
construct continua of higher than second order, continua,
that is, whose elements are, so to speak, compacted
more closely together than in the case of the Grand
Continuum; and what would be the bearing of such a
higher continuum upon the sort of phenomena above
indicated? The first part of the question is discussed by
Paul Du Bois-Reymond. The works of this author are
not easy to read. The student may be referred to G. H.
Hardy's Orders of Infinity: the In/initar-calcul of Paul