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lectures in Berlin (1859). The equivalence of the neighbourhood
definition and the sequential definition was established by Cantor in
1871 and the sequential definition of continuity was proposed by
Cantor and Heine in 1872. It was Weierstrass who really paved the way
for themodrn study of continuous functions by his construction of an
everywhere continuous and nowhere differentiable function, which
appeared in print in 1874. Until Riemann and Weierstrass,
considerations of continuity had not been clearly distinguished from
those of differentiability. In fact, Bolzano had found an everywhere
continuous but nowhere differentiable function in 1834, but was not
able to publish. In 1851 Riemann proved that a continuous function
was bounded in a neighbourhood of one of its points.
The use of 'lim' when referring to limits dates back to L'Huilier in
1786. Cauchy, in speaking of limits as x tends to 0, said that lim sin x
had onevalu, lim 1/ x had two values, and lim sin 1/ x had an infinity of
values. The study of limits from above and from below had been
stimulated by considerations of Fourier series and in 1837 Dirichlet
proposed the notation f ( a 0) and f ( a 0) for lim
f ( x ) and
lim
f ( x ), and his notation was used by many authors through the
nineteenth century. Weierstrass wrote lim
f ( x ), and thereplacing of
the equals sign by an arrow, which is now universal, comes from a
suggestion of the British mathematician J. G. Leathem in 1905.
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