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this was a theorem which required analytical proof. He provided the
first formal definition of a continuous function, showed that if a
continuous function was positiveat a point it must bepositivein a
neighbourhood of that point and likewise if it is negative at a point it
must be negative in a neighbourhood of that point. Taking the function
to be positive at the upper end of the domain and negative at the lower
end he claimed that the set of points of thedomain at which the
function was negative was bounded above and so must have a least
upper bound. At this least upper bound only a zero value for the
function is not contradictory. Bolzano proved the least upper bound
property using a Cauchy sequence, though not by that name (!), derived
from repeated bisections. His proof that Cauchy sequences converge
was incomplete because he thought it followed from the impossibility of
the convergence of such a sequence to two distinct limits. Perhaps
independently of Bolzano, Cauchy also produced a formal definition of
continuity in 1821 and the proof of the Intermediate Value Theorem
that we constructed in this chapter was essentially his. He repeatedly
divided his domain into
m
equal parts and then selected one part on
which a change of sign had taken place. We took
m
2. Cauchy
assumed, without proof, that monotonic bounded sequences were
convergent.
The theorem that continuous functions on a closed interval were
bounded and attained their bounds had been familiar from the
seventeenth century but under much stronger differentiability
conditions. The integral of Riemann's discontinuous function (invented
1854, but published 1867) provided an example that distinguished
clearly between continuous and differentiable functions for the first
time. It was Weierstrass in his lectures in Berlin in 1861 who first
proved that a continuous function on a closed interval was bounded
and attained its bounds. Weierstrass called this his 'Principal theorem'.
Weierstrass aMrmed the importance of considering continuous but not
necessarily differentiable functions by his approximation theorem that
every continuous function is the uniform limit of a sequence of
polynomials.
The distinction between continuity at a point and continuity on an
interval was not clear in Cauchy's work. In his proof of the integrability
of continuous functions (1823), Cauchy assumed continuity but used
uniform continuity. Dirichlet proved that a function which was
continuous on a closed interval was uniformly continuous on that
interval in his lectures in Berlin in 1854. The first proof of this result in
a Weierstassian context was published by E. Heine in 1872.
